Stochastic Subsurface Data Integration Assessing Aquifer Parameter and Boundary Condition Uncertainty

Size: px
Start display at page:

Download "Stochastic Subsurface Data Integration Assessing Aquifer Parameter and Boundary Condition Uncertainty"

Transcription

1 Journl of Hyrology mnusript No. (will e inserte y the eitor) Stohsti Susurfe Dt Integrtion Assessing Aquifer Prmeter n Bounry Conition Unertinty Dongong Wng Ye Zhng Reeive: te / Aepte: te Astrt This reserh suessfully extens eterministi, physilly-se inverse theory tht is ple of simultneous prmeter n ounry onition estimtion to unertinty quntifition in inverting stey-stte grounwter flow in two-imensionl quifer with fies heterogeneity. Using fies n ynmi flow mesurements smple t wells s oservtions, stohsti susurfe t integrtion tehnique is propose: (1) Sequentil Initor Simultion integrtes fies t y hrterizing its geosttistil prmeters (experimentl iretionl vriogrms n smple fies proportions) to generte orrelte fies moels; (2) for eh fies moel, hyruli onutivities n flow fiel (inluing the unknown ounry onitions) re estimte vi iret inversion metho; (3) unertinty in inversion, inluing oth unertinties of the estimte hyruli onutivities n the flow fiel, is evlute y ssessing the inversion outome for ll fies moels. To test the propose integrtion tehnique, referene forwr moel provies oth fies hrteriztion n ynmi mesurements t inresing smpling ensities (i.e., t quntity) n mesurement errors (i.e., t qulity). Vi smoothing n gri orsening, lterntive hyruli onutivity prmeteriztion is lso teste in inversion. Unertinty in the estimte onutivities n ounry onitions is then quntifie ginst the referene moel to evlute the qulity of inversion. Results suggest tht for the rnges of teste vrition in t quntity, qulity, n inverse onutivity prmeteriztion, (1) t quntity hs the strongest impt on oth inversion ury n preision; (2) t qulity influenes inversion ury; (3) inverse prmeteriztion hs the wekest influene on inversion s long s the overll fies pttern is pture (i.e., suffiient t quntity). A lne n thus e hieve etween prmeteriztion, omputtionl effiieny, n inversion performne. For Dongong Wng Deprtment of Geologil Sienes, University of North Crolin, Chpel Hill, North Crolin, USA E-mil: wng1986@live.un.eu Ye Zhng 1 University Ave., University of Wyoming, Lrmie, Wyoming, USA E-mil: yzhng9@uwyo.eu

2 2 Dongong Wng, Ye Zhng the heterogeneity pttern investigte herein, y efining n eptle mrgin of unertinty for either onutivity or flow fiel estimtion, optiml well sping in reltion to the hrteristi length of heterogeneity n e etermine uner unknown ounry onitions. Finlly, inversion omin shoul e losely efine y the mesurement lotions in orer to minimize extrpoltion errors. Keywors Hyrogeologil Moeling Unertinty quntifition Physillyse inversion Bounry onitions 1 Introution Prmeterizing hyruli onutivities (Ks) of n quifer moel is hllenging topi in mny susurfe investigtions. Grounwter moeling often suffers from signifint unertinty in epiting oth the K vlues n its sptil istriution in the susurfe, s extensive rilling n smpling to ollet hyrogeologil t is imprtil. However, oth quifer mngement n susurfe ontminnt lenup opertions require the evelopment of urte n ppropritely prmeterize grounwter moels. Deision-mking further requires sientifilly informe ssessment of the unertinty in the moel preitions, whih often rises from the unertinty in inferring susurfe hyruli onutivities n the ssoite flow fiel. In nturl quifers, onutivity istriution is heterogeneous t multiple sles n its iret mesurement often suffers from the well-known sle effet, i.e., Ks mesure y ifferent instruments hnge with the support volumes eing teste [1][2][3]. On the other hn, y eveloping numeril quifer moel, Ks of the moel n e lirte using inverse theory se on mesurements of the hyrogeologil stte vriles (e.g., hyruli hes, flow rtes). In prtiulr, inverse methos hve een evelope to not only filitte prmeter estimtion, ut lso to quntify the ssoite estimtion unertinty, whih n le to n ssessment of the preition unertinty when moels re use for eision-mking. Generlly, inverse methos in hyrogeology n e tegorize into iniret n iret methos. With the iniret inversion metho, n ojetive funtion, typilly efine s mismth etween the mesurement t n the orresponing moel simulte vlues, is minimize [4]. During inversion, prmeters re upte itertively y running forwr moel tht provies link etween the prmeters n the t. The upte prmeters then le to n upte forwr moel, with whih the ojetive funtion is reevlute. This proess is repete until user-efine fitting riterion is rehe. The iniret metho is foun flexile n effiient in lirting mny susurfe moels, lthough numer of issues hve een ientifie in its pplitions. Among them, ill-poseness is well-known prolem, whih is mnifeste y instility (sensitivity of the estimte prmeters to smll hnges in the oservtion t n their errors), nonuniqueness (more thn one set of prmeters n lirte the moel, with results epening on the strting points in the prmeter spe), n filure to onverge to resonle solution. Beuse forwr moel is neee to optimize the ojetive funtion, ounry onitions (BC) of the moel re either ssume known, or re lirte s prt of the inversion proess. However, BC of quifers re often unknown or unertin. As emonstrte y [5], ifferent omintions

3 Stohsti Dt Integrtion 3 of prmeters n BC n le to the sme ojetive funtion vlues, thus results of iniret inversion my eome non-unique. (In trnsient prolems, oth quifer initil n BC re unknown this topi is resse elsewhere [6] n is not evlute in this work.) For quifer prmeter estimtion, iret inversion methos hve lso een explore owing to their mthemtil strightforwrness n omputtionl effiieny. However, iret methos suffer from issues suh s high sensitivity to oservtion errors t stremline lotions [7][8][9], stringent error ouns on mesurements [1], reue ury with inresing prolem imensions [11], et. As reviewe in [12], these methos lso simulte the forwr moel requiring the speifition of BC, they therefore n suffer from instility n non-uniqueness in inversion n re not ommonly use in grounwter moel lirtion. Reently, se on the Stress Trjetories Element Metho evelope for solving ertin geophysil n flui mehnis prolems [13], new iret metho ws propose for stey-stte quifer inversion tht llows the simultneous estimtion of quifer onutivities, soures/sinks (e.g., rel rehrge, pumping n injetion), n stte vriles (e.g., hyruli he n Dry flux) [5][14][15][16]. Beuse the stte vriles re solutions of the inversion, the unknown quifer ounry onitions n thus e estimte. The iret metho honors the physis of flow using ontinuity priniples, while it iretly inorportes noisy oserve t t the mesurement lotions in single step, without using forwr simultions n itertive prmeter uptes to optimize ny ojetive funtion. Beuse it oes not simulte the forwr moel, the iret metho is omputtionlly effiient, n it utilizes n inversion gri with highly flexile isretiztion n prmeteriztion. However, previous works hve severl limittions tht restrit the metho to solving simple prolems with smll gri sizes. First of ll, onutivity istriution n ptterns were ssume to e known when formulting the inversion equtions. In rel quifers, K istriutions re typilly unknown or highly unertin. On the other hn, quifer hrteriztion n provie informtion on K istriution t orehole lotions. Seonly, inversion equtions hve so fr een evelope n teste for smll prolems: the mximum size of the inversion gri teste is (two-imensionl prolems) or (threeimensionl prolems), whih limits the inversion to resolving only orse heterogeneity fetures in the K moel. Conutivity in nturl quifers often exhiits irregulr sptil vrition, whih will require the evelopment of refine inversion gri with more gri ells. Thirly, euse the K ptterns were eterministi n ssume fully known, inversion outomes, inluing oth the estimte K vlues n the flow fiel, were eterministi. Unertinty in inversion thus oul not e quntifie. To emonstrte wier ppliility of the new iret metho, the ove issues nee to e resse y: (1) inorporting iret or iniret informtion out quifer heterogeneity, e.g., fies oservtions t orehole lotions; (2) inverting lrger gri sizes to etter resolve etile heterogeneities; (3) ounting for unertinty in inversion, inluing oth the unertinty of the estimte prmeters n the flow fiel. For stey-stte quifer flows, this reserh ims to ress the ove issues y eveloping n testing stohsti susurfe integrtion tehnique tht omines geosttistis (i.e., stti t integrtion) with iret inversion (i.e., ynmi t integrtion) into single estimtion n unertinty nlysis frmework. The previous inversion will thus e extene to inlue quifer pro-

4 4 Dongong Wng, Ye Zhng lems with relisti K heterogeneity, while the ssoite estimtion unertinty, inluing the unertinty in the inferre flow fiel, n e quntifie. To evlute the ppliility of the integrtion tehnique to fiel onitions with limite n noisy oservtions, this reserh first investigtes the impt of t quntity (i.e., numer of smpling wells) n qulity (i.e., mgnitue of mesurement errors) on the ury n stility of inversion, with results leing to set of smpling strtegies tht n rener the propose integrtion more ost-effetive. In prtiulr, y efining n eptle mrgin of unertinty in either K or flow fiel estimtion, optiml well sping in reltion to the hrteristi length of heterogeneity n e reommene. Here, the term optiml mens suffiient stti n ynmi mesurements n e otine from fewest wells (or lrgest well sping) tht n still provie onstrins to inversion, resulting in n eptle level of estimtion unertinty. Moreover, euse inverse K prmeteriztion is flexile n, s emonstrte y previous upsling stuies [17], high-resolution K heterogeneity is not inispensle for preiting ulk flow in quifers, this reserh will prmeterize the inversion gri with heterogeneity t vrying egrees of smoothness. In prtiulr, y orsening the inversion gir, lne etween omputtionl effiieny n estimtion ury/unertinty is explore. In the reminer of this rtile, the stohsti integrtion tehnique is introue first, followe y esription of syntheti forwr moel tht will e use s referene moel. The smpling esign is isusse, whih will provie inversion with fies n hyrogeologil t orehole lotions. Stti t integrtion vi geosttistil simultion is expline riefly, efore ynmi t integrtion vi the iret metho is introue. Beuse lrger inverse prolems re solve, mtrix onitioning tehniques re opte to improve the omputtionl effiieny of inversion. The results re presente in whih the effets of t qulity, quntity, n inverse prmeteriztion on inversion ury n unertinty re evlute. Finlly, implitions of these results for fiel implementtion of the propose integrtion tehnique is isusse n future reserh inite. 2 Metho In this work, stohsti integrtion tehnique is propose, onsisting of 3 steps: (1) Sequentil Initor Simultion (SIS) integrtes the fies t (i.e., experimentl iretionl vriogrms n smple fies proportions ompute from orehole mesurements) to generte rnom moels of orrelte fies; (2) for eh fies moel, hyruli onutivities n flow fiel (inluing the unknown BC) re estimte vi iret inversion; (3) unertinty in inversion inluing unertinties of the estimte Ks n the flow fiel is evlute y ssessing the outomes for ll fies moels. Beuse forwr moel is use s referene moel to test the qulity of inversion (i.e., estimtion ury n preision), elow, the forwr moel is esrie first. 2.1 Forwr (Referene) Moel To test n verify the propose integrtion tehnique, referene forwr moel, representing two-imensionl syntheti quifer trnset, ws rete with known

5 Stohsti Dt Integrtion 5 Fig. 1 The referene moel flow fiel ontining two fies (rk gry: silty sn with K1; light gry: len sn with K2). Given the speifie BC n fies Ks, hyruli he istriution is lso shown. Four ounry segments re lele: - enotes the inflow ounry, n - enotes the outflow ounry. fies ptterns n K vlues (Fig. 1). The grounwter flow eqution of the forwr moel, representing onfine quifer without soures/sinks, n e written s: (q) = (1) q = K(x, z) h (x, z) Ω where Ω is the sptil omin (ssume to e the sme for the forwr moel n for inversion), x n z represent the horizontl n vertil xes in Crtesin oorinte, is the grient opertor, h is hyruli he, q is Dry flux, n K(x, z) enotes lolly isotropi hyruli onutivity whih is prmeterize s fies in this work. Note tht though prior reserh hs resse inversion with soure/sink effets [14][16], this topi is eyon the sope of the urrent stuy n will e left for nother tretment. To rive stey-stte flow through the quifer, no-flow ounries re speifie to the moel top n ottom ounries, while onstnt hyruli hes re speifie on the left n right sies of the moel, i.e., feet n feet, respetively (Fig. 1). The mp of fies K is opte from [18], onsisting of 1 pixels in oth the horizontl n vertil iretions. The fies (true) onutivities were set to K 1 = 1 ft/yr (silty sn) n K 2 = 1 ft/yr (len sn) oring to lithology type [19]. Note tht the horizontl fies orreltion rnge of the len sn (light grey regions in Fig. 1) is pproximtely the lterl omin prolem size, i.e., 1 pixels. The forwr moel is isretize with uniform fine gri (N x = 5 n N z = 5; eh fies pixel is thus represente y 5 5 gri ells) n is solve with the finite ifferene metho (FDM) with MODFLOW [2]. (The forwr moel is lso referre to herein s the FDM.) Mss lne reports from MOD- FLOW suggest negligile errors on the orer of 1 7, thus the simulte flow fiel is onsiere error-free.

6 6 Dongong Wng, Ye Zhng GTM With Smple Hyruli He Lotions (12Wells) he lotion 2 2 GTM with Smple Flux Lotions (12Wells) flux lotion 4 4 Rows 6 Rows Columns Columns Fig. 2 Lotions of 12 smpling wells ple uniformly ross the quifer trnset. Along eh well, he n flux smpling points re shown. The referene fies mp (1 y 1 pixels) is shown in the kgroun. To provie the mesurements for inversion, set of syntheti oservtion wells, eh with with of one FDM gri ell (1/5 pixel with in the fies mp), ws smple for fies, hyruli hes, n point-sle Dry fluxes. Fies ws smple long the full length of eh well, representing orehole mesurements. Hyruli he n flux smplings were me t isrete lotions orresponing to multilevel onfigurtion, e.g., 1 hyruli he n 5 flux mesurements per well. A shemti igrm with 12 uniformly spe smpling wells is shown in Fig. 2. Moreover, if the lterl omin size is use to pproximte the horizontl fies orreltion rnge (λ H ), the lterl well sping of Fig. 2 is thus 1/12 of λ H. As prt of the nlysis on smpling ensity, the numer of smpling wells will e reue, i.e., 6 wells (well sping is 1/6 of λ H ) n 3 wells (well sping λ H well sping. is 1/3 of λ H ). A horizontl smpling inex (SI) n e efine s: For the ifferent smpling shemes, SI = 12 (12 wells), 6 (6 wells), n 3 (3 wells). A lrger SI vlue suggests higher smpling ensity, with orresponingly smller well sping in reltion to the lterl hrteristi length of the fies. Furthermore, twie n four times s mny mesurements were smple when SI is 12, ompre to when SI=6 n SI=3, respetively. For eh smpling ensity, the referene moel fies istriution, the fies Ks, n flui flow BC will e lter reovere using the propose stohsti integrtion tehnique. 2.2 Stti Dt Integrtion Initor geosttistis is use to ssimilte the oserve fies long the welloles [21]. Given the oe fies t (e.g., for silty sn, 1 for len sn), experimentl vriogrms re ompute long the horizonl n vertil iretions. These vriogrms re fitte with n exponentil moel, whih is foun to est pture the fies orreltion struture. When 12 wells were smple, Fig. 3 (top row) presents the experimentl fies vriogrms for len sn n the fitte

7 Stohsti Dt Integrtion 7 moels, whih re lso presente elow: γ( l ) = { [1 e ( lx 6 ) ] [1 e ( lz 4 ) ] (2) where γ is vriogrm n l is lg istne in pixels (l x is horizontl lg; l z is vertil lg). In fitting the moel, geometri nisotropy is ssume, thus the moele horizontl sill (not rehe in Fig. 3; top row) is the sme s the moele vertil sill. The fitte horizontl fies orreltion rnge is 15 pixels, whih is greter thn the fitte vertil fies orreltion rnge ( 15 pixels). Both prmeters re onsistent with the verge lterl extent n thikness of strtifition s exhiite y len sn (Fig. 1). Given the fitte vriogrm moels n the smple initor histogrms, Sequentil Initor Simultion (SIS) ws use to generte 1 fies reliztions, whih were onitione to the oserve wellore fies (Fig. 4). Eh fies reliztion hs the sme resolution s the originl fies mp, i.e., 1 1 pixels. When well ensity is hnge, the ove nlysis, inluing oth vriogrm moeling n SIS, is repete. Fig. 3 lso presents the vriogrm moels fitte when fewer wells were smple: not only will the fitte moels e less urte, the numer of onitioning fies t for SIS is signifintly reue. As result, s the numer of wells is reue, the SIS reliztion of the fies hs eome less urte (Fig. 4). For given smpling ensity, 1 fies reliztions will e rete. Using iret inversion, 1 flow fiels n 1 sets of fies K vlues will e etermine, one set for eh reliztion. To rive inversion, ynmi oservtions (i.e., hyrogeologil t) will e smple from the sme wells in the forwr moel.

8 8 Dongong Wng, Ye Zhng horizontl vriogrm (12 wells) vertil vriogrm (12 wells) experimentl vriogrm lg istne horizontl vriogrm (6 wells) experimentl vriogrm lg istne vertil vriogrm (6 wells) experimentl vriogrm experimentl vriogrm lg istne horizontl vriogrm (3 wells) lg istne experimentl vriogrm experimentl vriogrm lg istne vertil vriogrm (3 wells) lg istne Fig. 3 Vriogrm nlysis of the len sn fies when ifferent numer of wells ws smple. The otte lines enote the experimentl vriogrms n the smooth urves re the fitte exponentil moels. To eliminte ege effet, the mximum lg in oth horizontl n vertil iretions is hlf of the orresponing omin length sles, i.e., 5 pixels. 2.3 Dynmi Dt Integrtion For given fies reliztion, whih elinetes the K istriution, this reserh opte the iret metho of [5] to estimte fies K vlues n he istriution throughout the omin. To pply the iret metho, set of funmentl solutions of inversion is fitte lolly to the oserve ynmi t, while flow ontinuity is honore over ll inversion gri ells. The funmentl solutions of inversion re erive y solving the stey-stte grounwter flow eqution to otin set of lol nlytil solutions ssuming tht lol K of suomin or single inversion gri ell is homogeneous. However, unlike [5], whose funmentl solutions yiele only single K vlue (i.e., rtios etween fies Ks were ssume known s set of prior informtion onstrints for inversion), the funmentl so-

9 Stohsti Dt Integrtion 9 SIS Reliztion (12 Wells) 2 Rows Columns SIS Reliztion (6 Wells) 2 Rows Columns SIS Reliztion (3 Wells) 2 Rows Columns Fig. 4 SIS reliztions given 12, 6, n 3 smpling wells. For eh smpling ensity, one fies reliztion (out of 1 reliztions) is shown.

10 1 Dongong Wng, Ye Zhng lutions hve een moifie to llow the simultneous estimtion of oth Ks of the referene moel (this pproh is extenle to ny numer of fies): h(x, z) = + 1 x + 2 z + 3 xz + 4 (x 2 z 2 ) q x (x, z) = K( z x) q z (x, z) = K( x 2 4 z) (x, z) Ω i (3) where h enotes the pproximte hyruli he, ( q x, q z ) enote the pproximte grounwter fluxes, i (i =,..., 4) enote set of oeffiients tht lolly efine these pproximte solutions, K is lol hyruli onutivity: K (K 1, K 2,...) of the fies, n Ω i is suomin of the prolem, here orresponing to n inversion gri ell. The ontinuity equtions, whih penlize the mismth etween the funmentl solutions t the interfe etween jent inversion gri ells, n e written s: δ(p j (x j, z j ) ɛ)(k 1 h(k) (x, z) K 1 h(l) (x, z)) =, (k,l) K 1 δ(p j (x j, z j ) ɛ)(k 2 h(k) (x, z) K 2 h(l) (x, z)) =, (k,l) K 2 δ(p j (x j, z j ) ɛ)(k m h(k) (x, z) K m h(l) (x, z)) =, (k) K 1, (l) K 2, m (1, 2) δ(p j (x j, z j ) ɛ)( q n (k) (x, z) q n (l) (x, z)) =, K (k) K (l) δ(p j (x j, z j ) ɛ)( q (k) t (x, z) q (l) t (x, z)) =, K (k) = K (l) δ(p j (x j, z j ) ɛ)( q n (k) (x, z) q n (l) (x, z)) =, K (k) = K (l) (4) where p j (x j, z j ) enotes the jth ollotion point, whih lies on the interfe etween gri ells (k) n (l), q n is norml flux t p j, q t is tngentil flux t p j, δ(p j (x j, z j ) ɛ) is Dir elt weighting funtion [5] tht smples the mismth etween the funmentl solutions t p j (x j, z j ). The reltion etween ( q n, q t ) n ( q x, q z ) n e etermine using the ngles etween the interfe n the glol oorinte xes. Inversion further stisfies set of t onstrints whih n e written s: δ(p t ɛ)(k m h(k) (x t, z t ) K m h o (x t, z t )) = m (1, 2) δ(p t ɛ)( q (k) n (x t, z t ) q o n(x t, z t )) = δ(p t ɛ)( q (k) t (x t, z t ) q o t (x t, z t )) = (5) where δ(p t ɛ) is the Dir elt weighting funtion, whih reflets onfiene in the oserve t (e.g., it n e inversely proportionl to the mesurement error vrine), (x t, z t ) enotes the lotion where n oserve tum ws smple, n h o, qn, o qt o re the oservtions, K m enotes the onutivity of the fies whih ontins the oservtions. Flux mesurements re use here to provie flow rte relte informtion for inversion, euse onutivity nnot e uniquely ientifie from hyruli he oservtions lone. If susurfe flow rte mesurements re ville, however, the flux onitioning equtions n e integrte to enfore onitioning y flow rtes [5]. As ws isusse in [15], mesurements of in-situ fluxes n flow rtes n e me with vriety of orehole n wter lne tehniques.

11 Stohsti Dt Integrtion 11 Bse on the propose funmentl solutions [Eqn. (3)], y ssemling the ontinuity [Eqn. (4)] n the t equtions [Eqn. (5)], system of liner equtions ws evelope: A x (6) where A is the oeffiient mtrix, is the right hn sie vetor, n x is the solution vetor, where x = [K 1, K 2, (k), (k) 1, (k) 2, (k) 3, (k) 4,...], k = 1, 2,..., M (numer of ells in the inversion gri). Previous reserh [14] suggests tht if ynmi oservtions re too few resulting in uner-etermine inversion systems of equtions, stility of inversion n suffer. In this work, suffiient numer of he n flux oservtions ws smple (Fig. 2), thus Eqn. (6) is over-etermine n is solve with lest-squres minimiztion tehnique s implemente y the LSQR lgorithm [22]. For smller prolems reporte in previous reserh, LSQR ws foun to e n effiient solver. However, prolems inverte here re lrger with mny more unknowns, for whih the spee of onvergene ws foun to e sensitive to the onition numer of A. To improve mtrix onitioning, preonitioning tehniques inluing mtrix sling n Gussin Noise Perturtion were use efore the mtrix solve [23]. Depening on the prolem, pre-onitioning hs improve the solver spee y up to 1 times (generlly, greter speeup is hieve with lrger mtries), without suffering ill-effets in inversion ury. After pre-onitioning, inversion of single reliztion with the SIS gri will tke 1 min. The hyruli he fiel n then e reovere pieewise from the oeffiients efining the funmentl solutions t eh gri ell; from the estimte Ks, Dry flux fiel n e similrly reovere. The hyruli he BC n e otine y smpling the pproprite h(x, z) t the ounry lotions. Similrly, flux BC n e otine y smpling the pproprite q x (x, z) n q z (x, z) t the ounries. In this work, the inverte BC re presente s hyruli he vlues. Beuse forwr moel is not solve, the iret metho n employ flexile gris with vrious prmeteriztions for representing onutivity. In generl, the SIS-generte fies reliztions exhiit smll-sle rtifts whih le to pthiness in the simulte fies istriutions (Fig. 4). This my in turn impt the qulity of inversion, s greter egree of pthiness will le to fewer ontinuity equtions (i.e., the numer of flux ontinuity onstrints will e reue). Three K prmeteriztions re teste in this work: (1) inversion iretly opts the noisy SIS prmeteriztion with n ientil gri (N x = 1, N z = 1); (2) inversion opts the sme SIS gri (N x = 1, N z = 1), ut the SIS-simulte fies re smoothe y simulte nneling (SA)[24]; (3) inversion opts orsene gri (N x = 5, N z = 5) whose fies prmeteriztion is rete y upsling the SA fiel. For ese of referene, these prmeteriztions re lele s SIS, SA, n orsene inversion gris (exmple of eh gri n e seen in [25]). For the ltter two prmeteriztions, SIS reliztions re first smoothe, n then orsene, efore ynmi integrtion is rrie out to estimte the K n flow fiel ensemles. For given smpling ensity, y ompring the inversion outomes using ll gris, the impt of inverse K prmeteriztion on the qulity of inversion is ssesse. Moreover, if reue resolution in the prmeteriztion oes not signifintly egre inversion performne (i.e., ury n unertinty), lower resolutions will e preferre ue to their greter omputtionl effiieny.

12 12 Dongong Wng, Ye Zhng 2.4 Unertinty Quntifition Unertinty in the propose stohsti integrtion erives from oth stti n ynmi t unertinty, whih omes from limite orehole smpling n mesurement errors (if impose). For fies smpling long oreholes, no mesurement errors re impose, i.e., oserve fies type is ssume error-free. For smpling the ynmi t (hes, fluxes) from the sme wells, rnom mesurement errors re impose. Beuse the FDM is the referene moel, mesurements smple from this moel re error-free, tht is, true hes or true fluxes. To impose errors, the true hes re orrupte y rnom noises: h m = h F DM ± δh, where h m is mesure he provie to inversion, h F DM is true he, n δh is hyruli he mesurement error. The highest δh is ±1% of the totl hyruli he vrition in the forwr moel, with n solute error up to ±1 feet. This is resonle onsiering tht moern mesurement tehniques n etermine wter levels with preision s fine s 1 m [26]. Flux mesurements, though lso menle to errors, re ssume error-free. The effet of imposing errors on fluxes is similr to imposing errors on hes, while imposing oth errors t the sme time n le to inversion outomes tht re iffiult to interpret, e.g., positive error in oserve he grient n e nele y negtive error in nery flux mesurement. Given given fies prmeteriztion (SIS, SA, or orsene), 1 inversion systems of equtions were ssemle n solve, resulting in set of ensemle solutions inluing the inverte Ks, flow fiels (i.e., hes n fluxes), n BCs. The ury of eh solution n e ssesse y ompring the estimte prmeters n flow fiels to those of the referene moel. Beuse given flow fiel n e uniquely etermine from K istriution n BC, only the estimte Ks n the re ompre to those of the forwr moel. The inverte ensemle flow fiels will not e presente. Unertinty in the inverte onutivity of eh fies is evlute y set of ensemle error sttistis or performne metris: E[K inv ] K true ε K = 1% σ K = K true K inv mx K inv min 4K true 1% where K inv is the inverte fies onutivity, K true is the orresponing true onutivity, n E[ ] is expettion, tken here s rithmeti men. E[K inv ] is thus the ensemle men of the inverte fies onutivity. Kmx inv n Kmin inv enote the mximum n minimum inverte fies onutivity from the ensemle. To evlute the unertinty in the inverte BC, two performne metris were opte: n ε BC = (1/n) i=1 σ BC = n (1/n) i=1 (E[h inv i ( (h mx i ] h true h true i h min 4h true i i ) i ) 1% ) 2 1% where h inv i is the inverte hyruli he t ounry ell lotion i (n is the numer of ells long the omin ounry), h true i represents the true he t the (7) (8)

13 Stohsti Dt Integrtion 13 sme lotion, h mx i n h min i enote the mximum n minimum inverte hes t lotion i from the ensemle solution, respetively, n E[h inv i ] is the ensemle men of the reovere he t lotion i. Note tht y summing n verging over ll ounry ells, ε BC n σ BC represent the verge BC estimtion error n spre surrouning the ensemle men. In the following, inversion ury is efine y ε K n ε BC : higher vlues inite is in the ensemle men, thus poorer ury [4]. On the other hn, σ K n σ BC reflet the spre of the ensemle from the true prmeters n flow fiel, respetively, n re thus linke to the preision of inversion [4]. Higher preision is reflete y lower σ K n σ BC, n thus lower estimtion unertinty. The qulity of inversion is onsiere high if ε n σ re oth smll for onutivity estimtion n BC reovery. 3 Results In the unertinty stuy, the ury n preision of the ensemle inverse solutions re exmine when mesurement quntity, qulity, n inverse prmeteriztion re vrie. (1) Dt quntity ws vrie y reuing the numer of smpling wells from 12 to 6, n then to 3 wells, from whih stti n ynmi t were smple from the FDM. The oserve hyruli hes were then sujet to inresing mesurement errors (±1, ±2, ±5, ±1%), while inverse prmeteriztion opte the SIS (noisy) fies reliztions with 1 1 gri ells. (2) From the ove, t qulity ws exmine losely to evlute the effet of mesurement errors on inversion. (3) Using error-free mesurements from the 12 wells, inverse prmeteriztion ws then vrie, i.e., ynmi integrtion ws performe using the sme oserve hes n fluxes, ut with fies ptterns emoie y the SIS, SA, n orsene gris. (4) A o-effet stuy ws rrie out vrying two ftors t the sme time. In the following, units of the relevnt quntities re: K in ft/yr (1 ft/yr =.35 m/yr), q in ft/yr (1 ft/yr =.35 m/yr), h in ft (1 ft =.35 m), flow rte in ft 3 /yr (1 ft 3 /yr =.28 m 3 /yr). Alterntively, onsistent set of units n e ssume n ll unit informtion n e remove [27]. 3.1 Dt Quntity When 12 wells were smple from the forwr moel, stti n ynmi t from these wells were provie to the propose t integrtion: (1) given the oserve fies t the wells, vriogrm moel ws fitte, with whih 1 fies reliztions were generte with SIS; (2) for eh reliztion, ynmi t, smple t the sme wells, were inverte to otin fies Ks n the ssoite flow fiel; (3) for ll reliztions, n ensemle of inversion outomes ws generte, n Eqn.(7) n Eqn.(8) were use to ssess the qulity of inversion. The ynmi t were initilly error-free, efore inresing mesurement errors ±1, ±2, ±5, ±1% were impose (for the 5 levels of mesurement errors teste, 5 inversions were performe). When the smpling ensity ws reue to 6, n then to 3, the ove steps were repete (n itionl 1, inversions were performe).

14 14 Dongong Wng, Ye Zhng Reliztions wells 6 wells 3 wells Inverte K1 Histogrm (SIS Error Free) Vlue Inverte K2 Histogrm (SIS Error Free) Reliztions wells 6 wells 3 wells Vlue Fig. 5 Inverte istriutions of K1 n K2 otine with ifferent smpling ensities when mesurements were error-free. For eh fies, rrow points to the true onutivity vlue. Inverse prmeteriztion is SIS. The results for onutivity estimtion re summrize in Tle 1. Severl oservtions n e me: (1) for given mgnitue of the mesurement error, the qulity of inversion is highest when 12 wells were smple n lowest when 3 wells were smple (Fig. 5). Clerly, when smpling ensity is reue, fewer stti n ynmi t re ville for the integrtion n, oringly, inversion outome worsens. (2) for given smpling ensity, the qulity of inversion worsens with inresing mesurement errors (Tle 1). When the oserve hes were impose with ±1% errors, onutivity estimtion error eomes very high suggesting strong ises: when 12 wells were smple, ε K is 76% for K1 n 27% for K2; when 6 wells were smple, ε K is 12% for K1 n 4% for K2; when 3 wells were smple, ε K is 156% for K1 n 34% for K2. When 6 n 3 wells were smple, E[K1] even eomes negtive. This ours when he grients yiel the wrong sign ue to noisy mesurements, ut the orret flux mesurements were still use in inversion. (3) n optiml smpling ensity for urte K estimtion epens on the mgnitue of the mesurement error (tht is expete in fiel prolem) n n eptle level of estimtion error (tht is user efine). For exmple, if

15 Stohsti Dt Integrtion 15 ±2% is onsiere relisti mesurement error (solute mesurement error is ±2 ft), 3 wells (SI=3) will provie suffiient ury for inversion if K estimtion error of 12% is onsiere eptle. However, for the sme level of mesurement error, 6 wells (SI=6) will e neee if K estimtion error of 5% is onsiere eptle. (4) n optiml smpling ensity for preise K estimtion lso epens on the mgnitue of the mesurement error n n eptle level of estimtion error. For exmple, if ±2% is onsiere relisti mesurement error, 6 wells will provie suffiient preision if σ K of 7% is onsiere eptle. However, 12 wells will e neee if σ K of 5% is onsiere eptle. The results for BC estimtion re summrize in Tle 2, where ensemle sttistis re ompile for 4 ounry segments (Fig. 1):,,, n. Along the sme segment, the inverte ounry hes re ompre to the true hes (Fig. 6 presents results given error-free mesurements). Similr to K estimtion, for given mesurement error, the qulity of BC estimtion (i.e., ury n preision) is highest when 12 wells were smple n lowest when 3 wells were smple. For given smpling ensity, the qulity of inversion worsens with inresing mesurement errors (more on this lter). For BC estimtion, the optiml numer of smpling wells lso epens on the mgnitue of the mesurement error n n eptle level of BC estimtion error. For eh ounry segment, istint ehvior in the ensemle sttistis is lso oserve. For exmple, he reovery long ifferent segments is sensitive to well ensity to ifferent egrees. Both ury n preision of he reovery long the inflow ounry ( ) re not signifintly ffete y the eresing well ensity (Fig. 6). However, inversion long the outflow ounry ( ) egres ppreily when the numer of smpling well is reue. Inversion qulity is sptilly non-uniform, espite the ft tht the smpling wells re uniform. This suggests the importne of ientifying optiml smpling lotions. To unerstn this issue, future work will omine inversion with sensitivity nlysis [4][28]. For given smpling ensity, hyruli he reovery long the inflow ( ) n outflow ( ) ounries re generlly less urte n preise, when ompre to he reovery long the moel top n ottom ounries (, ). This n e expline y extrpoltion of the funmentl solutions in the moel omins etween the inflow ounry n the first smpling well, n etween the outflow ounry n the lst smpling well (Fig. 2). Beuse no mesurements were me there, the inverse solution, whih is onitione t the well lotions, is extrpolte towrs the ounries. Inversion qulity long moel top n ottom is higher, euse oth ounries re losely onitione y well t (Fig. 2). This suggests tht (1) inversion omin shoul e efine y the well lotions to reue extrpoltion errors, n (2) smpling long n, in effet proviing (prtil) BC to inversion, will likely improve inversion t these ounry lotions. 3.2 Dt Qulity Results from the previous setion re reexmine to ssess the effet of mesurement errors on inversion. Agin, s reporte in Tle 1 n Tle 2, qulity of the inverte Ks n BCs egres with inresing he mesurement errors n eresing smpling ensity. Fig. 7 presents the istriutions of the estimte K1

16 16 Dongong Wng, Ye Zhng Tle 1 Inversion ury n preision for the estimte fies onutivities. SIS gri 12 wells 6 wells 3 wells Aury Preision Aury Preision Aury Preision K1 K2 K1 K2 K1 K2 K1 K2 K1 K2 K1 K2 Error ε(%) ε(%) σ(%) σ(%) ε(%) ε(%) σ(%) σ(%) ε(%) ε(%) σ(%) σ(%) ±1% ±2% ±5% ±1% Error-free Oserve Dt SIS SA Corsene Aury Preision Aury Preision Aury Preision K1 K2 K1 K2 K1 K2 K1 K2 K1 K2 K1 K2 Wells ε(%) ε(%) σ(%) σ(%) ε(%) ε(%) σ(%) σ(%) ε(%) ε(%) σ(%) σ(%) wells (Error-free) Aury Preision setion - setion - setion - setion - setion - setion - setion - setion - Gris ε(%) ε(%) ε(%) ε(%) σ(%) σ(%) σ(%) σ(%) SIS SA Corsene

17 Stohsti Dt Integrtion 17 Tle 2 Inversion ury n preision for the estimte hyruli he BC. The ol lphets enote ifferent ounry segments (Fig. 1). 12 wells (SIS gri) Aury Preision setion - setion - setion - setion - setion - setion - setion - setion - Error ε(%) ε(%) ε(%) ε(%) σ(%) σ(%) σ(%) σ(%) ±1% ±2% ±5% ±1% wells (SIS gri) Aury Preision setion - setion - setion - setion - setion - setion - setion - setion - Error ε(%) ε(%) ε(%) ε(%) σ(%) σ(%) σ(%) σ(%) ±1% ±2% ±5% ±1% wells (SIS gri) Aury Preision setion - setion - setion - setion - setion - setion - setion - setion - Error ε(%) ε(%) ε(%) ε(%) σ(%) σ(%) σ(%) σ(%) ±1% ±2% ±5% ±1%

18 18 Dongong Wng, Ye Zhng Hyruli Hes Hyruli Hes Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells SIS) Bounry Cell Numer Oserve hes vs. Reovere hes (Error free 6Wells SIS) Bounry Cell Numer Oserve hes vs. Reovere hes (Error free 3Wells SIS) Bounry Cell Numer Fig. 6 Inverte istriutions of ounry hes otine with ifferent smpling ensities when mesurements were error-free. The re lso shown. The ol lphets enote the ounry lotions (Fig. 1).

19 Stohsti Dt Integrtion 19 n K2 uner inresing mesurement errors. Compre to K2 (len sn), estimtion of K1 (silty sn) egres more quikly with inresing errors. For the mesurement errors of, ±1, ±2, ±5, n ±1% of the totl he vrition, ε K for K1 is., 3., 7., 29., 76.%, n for K2, it is.3, 1.3, 2.9, 1.3, 27.4%; σ for K1 is 3.8, 4.3, 4.8, 6.5, 7.3, n for K2, it is 1.2, 1.5, 1.5, 2.2, 2.5%. When mesurement errors re low, inversion performne in estimting oth onutivities is similr, ut when errors grow higher, K2 estimtion is more urte n preise. Thus, onutivity of low-k fies my eome more iffiult to ientify when signifint errors exist in mesurements. This effet my e ttriute to the ft tht flow in low-k fies is more sensitive to vrition in the hyruli he n its sptil grient. The inverte ounry hes uner inresing mesurement errors re presente in Fig. 8. When 12 wells were smple, inresing mesurement errors only les to slight egrtion of the inverte ounry hes. For exmple, when mesurement errors re progressively inrese, slightly inrese σ BC is oserve long the inflow ounry ( ). When 6 wells were smple, this egrtion, Inverte K1 Histogrm (SIS 12wells Inresing Errors) Reliztions error error ±1% error ±2% error ±5% error ±1% Vlue Inverte K2 Histogrm (SIS 12wells Inresing Errors) Reliztions error error ±1% error ±2% error ±5% error ±1% Vlue Fig. 7 Distriution of the estimte K for eh fies uner inresing mesurement errors. (top) inverte K1 histogrms; (ottom) inverte K2 histogrms. Arrows point to the true vlues.

20 2 Dongong Wng, Ye Zhng Bounry Cell Numer Bounry Cell Numer Bounry Cell Numer Bounry Cell Numer Bounry Cell Numer Bounry Cell Numer Bounry Cell Numer Bounry Cell Numer Bounry Cell Numer Bounry Cell Numer 35 4 Oserve hes vs. Reovere hes (Error±5% 3Wells SIS) 4 15 Bounry Cell Numer 1 Oserve hes vs. Reovere hes (Error±2% 3Wells SIS) 5 5 Oserve hes vs. Reovere hes (Error±1% 3Wells SIS) 4 Oserve hes vs. Reovere hes (Error±1% 6Wells SIS) 15 Bounry Cell Numer Hyruli Hes Hyruli Hes 1 Oserve hes vs. Reovere hes (Error±1% 12Wells SIS) Oserve hes vs. Reovere hes (Error±5% 6Wells SIS) 4 Hyruli Hes Hyruli Hes 5 Oserve hes vs. Reovere hes (Error±5% 12Wells SIS) 35 Oserve hes vs. Reovere hes (Error±2% 6Wells SIS) Hyruli Hes Hyruli Hes 15 Bounry Cell Numer Oserve hes vs. Reovere hes (Error±2% 12Wells SIS) 1 Oserve hes vs. Reovere hes (Error±1% 6Wells SIS) 5 Oserve hes vs. Reovere hes (Error free 3Wells SIS) Hyruli Hes Hyruli Hes Hyruli Hes Oserve hes vs. Reovere hes (Error±1% 12Wells SIS) Hyruli Hes Hyruli Hes Oserve hes vs. Reovere hes (Error free 6Wells SIS) Hyruli Hes Bounry Cell Numer 35 4 Oserve hes vs. Reovere hes (Error±1% 3Wells SIS) Hyruli Hes Hyruli Hes Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells SIS) Bounry Cell Numer 35 4 Fig. 8 Reovere ounry he ensemles uner eresing smpling ensity: (first olumn) 12 wells, (seon olumn) 6 wells, (thir olumn) 3 wells. In eh olumn, ounry hes re inverte uner inresing mesurement errors: (first row) %, (seon row) ±1%, (thir row) ±2%, (fourth row) ±5%, (fifth row) ±1%. in terms of inversion ury n preision, is more pronoune. When 3 wells were smple, similr egree of egrtion s tht of the 6 wells is oserve. Overll, for the prolem investigte here, BC reovery is not very sensitive to the impose rnom mesurement errors when suffiient numer of wells (12 wells) were smple. On the other hn, for prtil sitution where mesurement errors n e quntifie (e.g., ±2%), optiml smple sping n gin e ientifie y setting eptle tolernes for εbc n σbc. 3.3 Inverse Prmeteriztion Given error-free mesurements from 12 wells, inversion outome is presente when fies is prmeterize lterntively with SIS, SA, n the orsene gris (Fig. 9, Fig. 1). When the SIS gri is use, men onutivity preition is extremely lose to the true vlue, while σk remins very smll. When the SA gri is use, oth ury n preision of the inversion suffer slightly. When the orsene gri is use, inversion qulity is omprly the worst. The verge CPU time for solving n inverse prolem ws 55 s (SIS), 45 s (SA), n 33 s (orsene), thus speeup ue to heterogeneity smoothing n further orsening is roun 18% n

21 Stohsti Dt Integrtion 21 4%, respetively. When BC reovery is ompre, however, inversion performne oes not vry signifintly with the vrition in prmeteriztions (Fig. 1). The ove finings suggest tht K estimtion is sensitive to its prmeteriztion (i.e., smoothness in heterogeneity), while hyruli he reovery is not. It is wellknown tht hyruli he preition is not sensitive to smll-sle K vrition euse the grounwter flow eqution ts s filter: the high wve numer omponents orresponing to smll-sle heterogeneity re generlly filtere out. Moreover, if 1% K estimtion error is onsiere eptle, inversion using the orsene gri is optiml, s the spee of solver with this smller gri is fster y pproximtely 4% (ompre to the SIS gri) n 27% (ompre to the SA gri). 3.4 A Co-Effet Stuy In the previous setions, inversion ury n preision were nlyze seprtely ginst hnging oservtion quntity, qulity, n inverse prmeteriztion. To Reliztions SIS SA Corsening Inverte K1 Histogrm (12wells Error Free) Vlue Inverte K2 Histogrm (12wells Error Free) Reliztions SIS SA Corsening Vlue Fig. 9 Distriution of the estimte K for eh fies uner ifferent inverse prmeteriztions. (top) inverte K1 histogrms; (ottom) inverte K2 histogrms. Arrows point to the true vlues.

22 22 Dongong Wng, Ye Zhng Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells SIS) Bounry Cell Numer Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells SA) Bounry Cell Numer Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells Corsening) Bounry Cell Numer Fig. 1 Reovere ensemles of ounry hes long ---- uner ifferent inverse prmeteriztions: (top) SIS, (mile) SA, (ottom) orsene gri. unerstn the reltive importne of eh ftor to the performne of inversion, o-effet nlysis ws onute. To etermine the reltive rnking, three seprte two-imensionl experiments vrying two ftors t time were esigne, i.e., t quntity versus prmeteriztion, t quntity versus t qulity, t qulity versus prmeteriztion. For exmple, when the o-effet of t quntity n prmeteriztion is of interest, three ifferent prmeteriztions were employe for given smpling ensity, resulting in 3 sets of stohsti inversions (i.e., ensemle solutions). For the 3 smpling ensities (12 versus 6 versus 3 wells), 9 inversion equtions were thus solve. The inversion outomes ensemle mens n stnr evition of K or BC estimtion, were then nlyze in light of the joint vrition of the two ftors. For given smpling ensity, this nlysis revels tht prmeteriztion with the orsene gri lwys les to less ury n lower preision in inversion, s expete. The sme n e si when fewer wells re

23 Stohsti Dt Integrtion 23 provie to inversion uner fixe prmeteriztion. To unerstn the reltive importne of eh ftor on inversion, mtries of the ensemle outomes n e rete n nlyze [25]. For the rnges of teste vrition in t quntity n inverse prmeteriztion, t quntity is foun to the ominnt ftor tht influenes oth the ury n preision of K n BC estimtions. Similr two-imensionl experiments were rrie out to evlute the reltive importne of t quntity versus t qulity, n t qulity versus prmeteriztion (n itionl 1,8 ensemle solutions were otine n nlyze). Detil out these experiments n the o-effet stuy is provie in [25], n will not e presente. Here, we summrize the insights gine from this nlysis. For the rnges of teste vrition in t quntity, qulity, n inverse prmeteriztion, t quntity plys the ominnt role in etermining the ury n preision of the inverte Ks. Dt qulity is importnt to the ury of K inversion, ut hs limite influenes on its preision. Compre to t quntity n qulity, the impt of inverse prmeteriztion on the qulity of inversion is generlly smller, suggesting the viility of using upsle prmeter fiels with orse gris to hieve greter omputtion effiieny. Tht is, resolution-ury tre-off exists for prmeter estimtion. For BC n thus flow fiel estimtion, the most importnt ftor tht influenes the qulity of inversion is t quntity, while t qulity n heterogeneity resolution hve lesser influenes. An, regrless of the omintions exmine, regions of the inversion omin ner the inflow n outflow ounries, when ompre to the interior regions, lwys orrespon to lower inversion ury n preision. This is result of extrpoltion errors s no mesurements exist long these ounry segments. This is onsistent with the erlier results with eterministi inversion [16]. 4 Conlusions In this reserh, stohsti susurfe t integrtion tehnique is propose y omining geosttistil simultion with iret inverse metho to filitte prmeter n flow fiel estimtion in stey-stte quifer inversion while quntifying the ssoite estimtion unertinty. The oservtion t re otine from set of smpling wells, n inlue stti t (i.e., fies types) n ynmi t (i.e., hyruli hes n flux mesurements). The stti t re ssume error-free, while inresing mesurement errors re impose onto the oserve hyruli hes. First, sequentil initor simultion is use to integrte the stti t, whih les to set of orrelte fies reliztions with ifferent hyruli onutivity istriutions. This fies ensemle is then provie to iret inversion whih utilizes the ynmi t smple from the sme wells to estimte set of ensemle prmeters n flow fiels. With these ensemle inversion outomes, the men (expete) prmeters n flow fiels n their estimtion unertinty n then e etermine. Beuse the ove stohsti integrtion requires ensemle inversions on mny fies reliztions, omputtionl effiieny is of interest. Vi smoothing n gri orsening, fies prmeteriztion y SIS n e moifie y reuing the resolution of the fies moel prior to inversion. The smoothing of the fies n improve mtrix onitioning in inversion, while smoothing omine

24 24 Dongong Wng, Ye Zhng with gri orsening n le to reue eqution size. Both pprohes n reue the omputtionl time neee to solve the inversion systems of equtions. To test the propose integrtion tehnique, referene forwr moel provies oth the fies hrteriztion n ynmi mesurements t inresing smpling ensities (i.e., t quntity) n mesurement errors (i.e., t qulity). Beuse flow fiel n e uniquely etermine if onutivities n BC re known, the estimte BC ensemle is exmine in lieu of the flow fiels. Unertinty in the estimte onutivities n BC is quntifie ginst the referene moel to evlute the qulity of inversion. Results suggest tht for the rnges of teste vrition in t quntity, qulity, n inverse prmeteriztion, (1) t quntity hs the gretest impt on oth inversion ury n inversion preision; (2) t qulity impts inversion ury; (3) inverse prmeteriztion hs the wekest influene on inversion s long s the overll fies pttern is pture (i.e., suffiient t quntity). A lne n thus e hieve etween prmeteriztion, omputtionl effiieny, n inversion performne. Moreover, for the heterogeneity investigte herein, y efining n eptle mrgin of unertinty for either onutivity or flow fiel estimtion, optiml well sping in reltion to the hrteristi length of heterogeneity n e etermine uner unknown quifer BC. Finlly, results of this stuy suggest tht inversion omin shoul e losely efine y the mesurement lotions in orer to minimize extrpoltion errors. Results n insights of this work n le to fiel implementtion of the propose stohsti integrtion tehnique for prolems with unknown informtion out quifer s ounry onitions. Along with prmeter estimtion, the integrtion tehnique n expliitly quntify ounry onitions inluing their men vlues n their unertinty, thus proviing powerful tool for hrterizing eep, t-poor environments whih re inresingly eing exploite for wste isposl (e.g., geologil ron storge n sequestrtion) s well s for energy proution (e.g., hyruli frking, geotherml, oil/gs) opertions. Using the integrtion tehnique, y ssuming resonle level of mesurement errors in the hyrogeologil t, well sping rnging from 1/3 to 1/6 of the lterl fies orreltion length n le to equte inversion outomes with eptle estimtion unertinties. Future work will exten the methos of this stuy to three-imensionl prolems, while stti t integrtion will inorporte not only iret orehole informtion ut other uxiliry t (e.g., seismi fies nlysis) tht n yiel itionl informtion of susurfe heterogeneity in etween wells. Aknowlegements This work is supporte y the University of Wyoming Shool of Energy Resoures Center for Funmentls of Susurfe Flow (WYDEQ49811ZHNG), n NSF CI- WATER (Cyerinfrstruture to Avne High Performne Wter Resoure Moeling), n NSF EPSCoR (EPS 12899). Referenes 1. Rihr L. Chmers, Jeffrey M. Yrus, n Kirk B. Hir. Petroleum Geosttistis for Nongeosttistiins. The Leing Ege, 19(5): ,. 2. C. Deutsh n P. Cokerhm. Prtil onsiertions in the ppliton of simulte nneling to stohsti simultion. Mthemtil Geology, 26(1):67 82, 1994.

25 Stohsti Dt Integrtion C. V. Deutsh n A. G. Journel. GSLIB: Geosttistil Softwre Lirry n User s Guie, Applie Geosttistis Series. Oxfor University Press, New York, NY 116, 2 eition, Bnt E.R. Hill M.C. Hrugh, A.W. n M.G. MDonl. MODFLOW-, the U.S. Geologil Survey moulr groun-wter moel User guie to moulriztion onepts n the Groun-Wter Flow Proess. U.S. Geologil Survey, Reston, Virgini, open-file report -92, 121 p. eition,. 5. M. C. Hill n C. R. Tieemn. Effetive Grounwter Moel Clirtion: With Anlysis of Dt, Sensitivities, Preitions, n Unertinty. Wiley-Intersiene, Hooken, N. J., J. Irs n A. N. Glyin. Stress Trjetories Element Metho for Stress Determintion from Disrete Dt on Prinipl Diretions. Eng. Anl. Bounry Elem., 34:423432, Jurj Irs n Ye Zhng. A Diret Metho of Prmeter Estimtion for Stey Stte Flow in Heterogeneous Aquifers with Unknown Bounry Conitions. Wter Resour. Res., 48:W9526, J. Jio n Y. Zhng. A metho se on Lol Approximte Solution (LAS) for Inverting Trnsient Flow in Heterogeneous Aquifers. Journl of Hyrology, 514: , J. Jio n Y. Zhng. Two-Dimensionl Inversion of Confine n Unonfine Aquifers uner Unknown Bounry Conitions. Avnes in Wter Resoures, 65:43 57, D. Kleineke. Use of Liner Progrming for Estimting Geohyrologi Prmeters of Grounwter Bsins. Wter Resour. Res., 7(2):367374, R. W. Nelson. In Ple Mesurement of Permeility in Heterogeneous Mei: 1. Theory of Propose Metho. J. Geophys. Res., 65: , R. W. Nelson. In Ple Mesurement of Permeility in Heterogeneous Mei: 2. Experimentl n Computtionl Consiertions. J. Geophys. Res., 66: , R. W. Nelson. In Ple Determintion of Permeility Distriution of Heterogeneous Porous Mei through Anlysis of Energy Dissiption. So. Pet. Eng. J., 3:3342, S. P. Neumn, A. Blttstein, M. Riv, D. M. Trtkovsky, A. Gugnini, n T. Ptk. Type urve interprettion of lte-time pumping test t in rnomly heterogeneous quifers. Wter Resoures Reserh, 43:W1421, C. C. Pige n M. A. Suners. Lsqr: An Algorithm for Sprse Liner Equtions n Sprse Lest Squres. Trns. Mth. Softwre, 8(1):43 71, V. E. A. Post n J. R. von Asmuth. Review: hyruli he mesurements new tehnologies, lssi pitflls. Hyrogeology Journl, 21:737 75, A. Sltelli, M. Rtto, T. Anres, F. Cmpolongo, J. Crioni, D. Gtelli, M. Sisn, n S. Trntol. Glol Sensitivity Anlysis: The Primer. Wiley. 18. X. Snhez-Vil, J. Crrer, n J. Girri. Sle effets in trnsmissivity. Journl of Hyrology, 183:1 22, V. Shevnin, O. Delgo-Rorguez, A. Moustov, n A. Ryjov. Estimtion of Hyruli Conutivity on Cly Content in Soil Determine from Resistivity Dt. Geofsi Internionl, 45(3):195 27, G. W. Stewrt n J. G. Sun. Mtrix Perturtion Theory. Aemi Press, 374 Pges, N.-Z. Sun. Inverse Prolems in Grounwter Moeling, Theory Appl. Trnsp. Porous Mei, vol. 6. Kluwer A., Dorreht, Netherlns, D. Wng. Physilly se stohsti inversion n prmeter unertinty ssessment on onfine quifer with highly slle prllel solver. Mster s thesis, University of Wyoming, W. W.-G. Yeh, Y. S. Yoon, n K. S. Lee. Aquifer Prmeter Ientifition with Kriging n Optimum Prmetriztion. Wter Resour. Res., 19(1):225233, Y. Zhng. Nonliner Inversion of n Unonfine Aquifer: Simultneous Estimtion of Heterogeneous Hyruli Conutivities, Rehrge Rtes, n Bounry Conitions. Trnsport in Porous Mei, 12: , Y. Zhng, J. Irs, n J. Jio. Three-Dimensionl Aquifer Inversion uner Unknown Bounry Conitions. Journl of Hyrology, 59: , Y. Zhng, B. Liu, n C. W. Gle. Homogeniztion of Hyruli Conutivity for Hierrhil Seimentry Deposits t Multiple Sles. Trnsport in Porous Mei, 87(3): , Y. Zhng, M. Person, n C. W. Gle. Representtive Hyruli Conutivity of Hyrogeologi Units: Insights from n Experimentl Strtigrphy. Journl of Hyrology, 339:65 78, 7.

Now we must transform the original model so we can use the new parameters. = S max. Recruits

Now we must transform the original model so we can use the new parameters. = S max. Recruits MODEL FOR VARIABLE RECRUITMENT (ontinue) Alterntive Prmeteriztions of the pwner-reruit Moels We n write ny moel in numerous ifferent ut equivlent forms. Uner ertin irumstnes it is onvenient to work with

More information

Lecture 6: Coding theory

Lecture 6: Coding theory Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those

More information

Bivariate drought analysis using entropy theory

Bivariate drought analysis using entropy theory Purue University Purue e-pus Symposium on Dt-Driven Approhes to Droughts Drought Reserh Inititive Network -3- Bivrite rought nlysis using entropy theory Zengho Ho exs A & M University - College Sttion,

More information

A Primer on Continuous-time Economic Dynamics

A Primer on Continuous-time Economic Dynamics Eonomis 205A Fll 2008 K Kletzer A Primer on Continuous-time Eonomi Dnmis A Liner Differentil Eqution Sstems (i) Simplest se We egin with the simple liner first-orer ifferentil eqution The generl solution

More information

CS 491G Combinatorial Optimization Lecture Notes

CS 491G Combinatorial Optimization Lecture Notes CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,

More information

MCH T 111 Handout Triangle Review Page 1 of 3

MCH T 111 Handout Triangle Review Page 1 of 3 Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

Edexcel Level 3 Advanced GCE in Mathematics (9MA0) Two-year Scheme of Work

Edexcel Level 3 Advanced GCE in Mathematics (9MA0) Two-year Scheme of Work Eexel Level 3 Avne GCE in Mthemtis (9MA0) Two-yer Sheme of Work Stuents stuying A Level Mthemtis will tke 3 ppers t the en of Yer 13 s inite elow. All stuents will stuy Pure, Sttistis n Mehnis. A level

More information

ANALYSIS AND MODELLING OF RAINFALL EVENTS

ANALYSIS AND MODELLING OF RAINFALL EVENTS Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F.

More information

Eigenvectors and Eigenvalues

Eigenvectors and Eigenvalues MTB 050 1 ORIGIN 1 Eigenvets n Eigenvlues This wksheet esries the lger use to lulte "prinipl" "hrteristi" iretions lle Eigenvets n the "prinipl" "hrteristi" vlues lle Eigenvlues ssoite with these iretions.

More information

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of: 22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)

More information

Lecture Notes No. 10

Lecture Notes No. 10 2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite

More information

Technology Mapping Method for Low Power Consumption and High Performance in General-Synchronous Framework

Technology Mapping Method for Low Power Consumption and High Performance in General-Synchronous Framework R-17 SASIMI 015 Proeeings Tehnology Mpping Metho for Low Power Consumption n High Performne in Generl-Synhronous Frmework Junki Kwguhi Yukihie Kohir Shool of Computer Siene, the University of Aizu Aizu-Wkmtsu

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs

Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if

More information

Momentum and Energy Review

Momentum and Energy Review Momentum n Energy Review Nme: Dte: 1. A 0.0600-kilogrm ll trveling t 60.0 meters per seon hits onrete wll. Wht spee must 0.0100-kilogrm ullet hve in orer to hit the wll with the sme mgnitue of momentum

More information

2.4 Theoretical Foundations

2.4 Theoretical Foundations 2 Progrmming Lnguge Syntx 2.4 Theoretil Fountions As note in the min text, snners n prsers re se on the finite utomt n pushown utomt tht form the ottom two levels of the Chomsky lnguge hierrhy. At eh level

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

Review Topic 14: Relationships between two numerical variables

Review Topic 14: Relationships between two numerical variables Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph

More information

CS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014

CS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014 S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown

More information

SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVEX STOCHASTIC PROCESSES ON THE CO-ORDINATES

SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVEX STOCHASTIC PROCESSES ON THE CO-ORDINATES Avne Mth Moels & Applitions Vol3 No 8 pp63-75 SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE STOCHASTIC PROCESSES ON THE CO-ORDINATES Nurgül Okur * Imt Işn Yusuf Ust 3 3 Giresun University Deprtment

More information

18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106

18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106 8. Problem Set Due Wenesy, Ot., t : p.m. in - Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the one-element, linerly epenent sets forme from these. (b) Wht re the two-element, linerly

More information

F / x everywhere in some domain containing R. Then, + ). (10.4.1)

F / x everywhere in some domain containing R. Then, + ). (10.4.1) 0.4 Green's theorem in the plne Double integrls over plne region my be trnsforme into line integrls over the bounry of the region n onversely. This is of prtil interest beuse it my simplify the evlution

More information

Conservation Law. Chapter Goal. 6.2 Theory

Conservation Law. Chapter Goal. 6.2 Theory Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the

More information

Lecture 2: Cayley Graphs

Lecture 2: Cayley Graphs Mth 137B Professor: Pri Brtlett Leture 2: Cyley Grphs Week 3 UCSB 2014 (Relevnt soure mteril: Setion VIII.1 of Bollos s Moern Grph Theory; 3.7 of Gosil n Royle s Algeri Grph Theory; vrious ppers I ve re

More information

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL P3.1 Kot Iwmur*, Hiroto Kitgw Jpn Meteorologil Ageny 1. INTRODUCTION Jpn Meteorologil Ageny

More information

Appendix A: HVAC Equipment Efficiency Tables

Appendix A: HVAC Equipment Efficiency Tables Appenix A: HVAC Equipment Effiieny Tles Figure A.1 Resientil Centrl Air Conitioner FEMP Effiieny Reommention Prout Type Reommene Level Best Aville 11.0 or more EER 14.6 EER Split Systems 13.0 or more SEER

More information

I 3 2 = I I 4 = 2A

I 3 2 = I I 4 = 2A ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents

More information

CSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4

CSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4 Am Blnk Leture 13 Winter 2016 CSE 332 CSE 332: Dt Astrtions Sorting Dt Astrtions QuikSort Cutoff 1 Where We Are 2 For smll n, the reursion is wste. The onstnts on quik/merge sort re higher thn the ones

More information

Lecture 8: Abstract Algebra

Lecture 8: Abstract Algebra Mth 94 Professor: Pri Brtlett Leture 8: Astrt Alger Week 8 UCSB 2015 This is the eighth week of the Mthemtis Sujet Test GRE prep ourse; here, we run very rough-n-tumle review of strt lger! As lwys, this

More information

Statistics in medicine

Statistics in medicine Sttistis in meiine Workshop 1: Sreening n ignosti test evlution Septemer 22, 2016 10:00 AM to 11:50 AM Hope 110 Ftm Shel, MD, MS, MPH, PhD Assistnt Professor Chroni Epiemiology Deprtment Yle Shool of Puli

More information

Linear Algebra Introduction

Linear Algebra Introduction Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +

More information

Engr354: Digital Logic Circuits

Engr354: Digital Logic Circuits Engr354: Digitl Logi Ciruits Chpter 4: Logi Optimiztion Curtis Nelson Logi Optimiztion In hpter 4 you will lern out: Synthesis of logi funtions; Anlysis of logi iruits; Tehniques for deriving minimum-ost

More information

MATH 122, Final Exam

MATH 122, Final Exam MATH, Finl Exm Winter Nme: Setion: You must show ll of your work on the exm pper, legily n in etil, to reeive reit. A formul sheet is tthe.. (7 pts eh) Evlute the following integrls. () 3x + x x Solution.

More information

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite! Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:

More information

Necessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 )

Necessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 ) Neessry n suient onitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deite to Professor Anne Penfol Street Astrt We give new lgorithm whih llows us

More information

TIME-VARYING AND NON-LINEAR DYNAMICAL SYSTEM IDENTIFICATION USING THE HILBERT TRANSFORM

TIME-VARYING AND NON-LINEAR DYNAMICAL SYSTEM IDENTIFICATION USING THE HILBERT TRANSFORM Proeeings of ASME VIB 5: th Biennil Conferene on Mehnil Virtion n Noise Septemer 4-8, 5 Long Beh, CA, USA DETC5-84644 TIME-VARYING AND NON-LINEAR DYNAMICAL SYSTEM IDENTIFICATION USING THE HILBERT TRANSFORM

More information

Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 3 : Interaction by Particle Exchange and QED. Recap

Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 3 : Interaction by Particle Exchange and QED. Recap Prtile Physis Mihelms Term 2011 Prof Mrk Thomson g X g X g g Hnout 3 : Intertion y Prtile Exhnge n QED Prof. M.A. Thomson Mihelms 2011 101 Rep Working towrs proper lultion of ey n sttering proesses lnitilly

More information

CH 17: Flexible Mechanical Elements

CH 17: Flexible Mechanical Elements CH 17: lexible Mehnil Elements lexible mehnil elements (belts, hins, ropes re use in onveying systems n to trnsmit power over long istnes (inste of using shfts n gers. The use of flexible elements simplifies

More information

Total score: /100 points

Total score: /100 points Points misse: Stuent's Nme: Totl sore: /100 points Est Tennessee Stte University Deprtment of Computer n Informtion Sienes CSCI 2710 (Trnoff) Disrete Strutures TEST 2 for Fll Semester, 2004 Re this efore

More information

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light.

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light. 1 This igrm represents the energy hnge tht ours when eletron in trnsition metl ion is exite y visile light. Give the eqution tht reltes the energy hnge ΔE to the Plnk onstnt, h, n the frequeny, v, of the

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

Let s divide up the interval [ ab, ] into n subintervals with the same length, so we have

Let s divide up the interval [ ab, ] into n subintervals with the same length, so we have III. INTEGRATION Eonomists seem muh more intereste in mrginl effets n ifferentition thn in integrtion. Integrtion is importnt for fining the epete vlue n vrine of rnom vriles, whih is use in eonometris

More information

Section 2.3. Matrix Inverses

Section 2.3. Matrix Inverses Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue

More information

Metaheuristics for the Asymmetric Hamiltonian Path Problem

Metaheuristics for the Asymmetric Hamiltonian Path Problem Metheuristis for the Asymmetri Hmiltonin Pth Prolem João Pero PEDROSO INESC - Porto n DCC - Fule e Ciênis, Universie o Porto, Portugl jpp@f.up.pt Astrt. One of the most importnt pplitions of the Asymmetri

More information

CARLETON UNIVERSITY. 1.0 Problems and Most Solutions, Sect B, 2005

CARLETON UNIVERSITY. 1.0 Problems and Most Solutions, Sect B, 2005 RLETON UNIVERSIT eprtment of Eletronis ELE 2607 Swithing iruits erury 28, 05; 0 pm.0 Prolems n Most Solutions, Set, 2005 Jn. 2, #8 n #0; Simplify, Prove Prolem. #8 Simplify + + + Reue to four letters (literls).

More information

Threshold and Above-Threshold Performance of Various Distributed Feedback Laser Diodes

Threshold and Above-Threshold Performance of Various Distributed Feedback Laser Diodes Threshol n Aove-Threshol Performne of Vrious Distriute Feek Lser Dioes C. F. Fernnes Instituto e Teleomunições, Instituto Superior Ténio, 1049-001 Liso, Portugl Astrt A omprtive nlysis of the threshol

More information

Implication Graphs and Logic Testing

Implication Graphs and Logic Testing Implition Grphs n Logi Testing Vishwni D. Agrwl Jmes J. Dnher Professor Dept. of ECE, Auurn University Auurn, AL 36849 vgrwl@eng.uurn.eu www.eng.uurn.eu/~vgrwl Joint reserh with: K. K. Dve, ATI Reserh,

More information

SOME COPLANAR POINTS IN TETRAHEDRON

SOME COPLANAR POINTS IN TETRAHEDRON Journl of Pure n Applie Mthemtis: Avnes n Applitions Volume 16, Numer 2, 2016, Pges 109-114 Aville t http://sientifivnes.o.in DOI: http://x.oi.org/10.18642/jpm_7100121752 SOME COPLANAR POINTS IN TETRAHEDRON

More information

University of Sioux Falls. MAT204/205 Calculus I/II

University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

More information

Algebra 2 Semester 1 Practice Final

Algebra 2 Semester 1 Practice Final Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

More information

CS 360 Exam 2 Fall 2014 Name

CS 360 Exam 2 Fall 2014 Name CS 360 Exm 2 Fll 2014 Nme 1. The lsses shown elow efine singly-linke list n stk. Write three ifferent O(n)-time versions of the reverse_print metho s speifie elow. Eh version of the metho shoul output

More information

Logic, Set Theory and Computability [M. Coppenbarger]

Logic, Set Theory and Computability [M. Coppenbarger] 14 Orer (Hnout) Definition 7-11: A reltion is qusi-orering (or preorer) if it is reflexive n trnsitive. A quisi-orering tht is symmetri is n equivlene reltion. A qusi-orering tht is nti-symmetri is n orer

More information

Section 2.1 Special Right Triangles

Section 2.1 Special Right Triangles Se..1 Speil Rigt Tringles 49 Te --90 Tringle Setion.1 Speil Rigt Tringles Te --90 tringle (or just 0-60-90) is so nme euse of its ngle mesures. Te lengts of te sies, toug, ve very speifi pttern to tem

More information

Common intervals of genomes. Mathieu Raffinot CNRS LIAFA

Common intervals of genomes. Mathieu Raffinot CNRS LIAFA Common intervls of genomes Mthieu Rffinot CNRS LIF Context: omprtive genomis. set of genomes prtilly/totlly nnotte Informtive group of genes or omins? Ex: COG tse Mny iffiulties! iology Wht re two similr

More information

Generalization of 2-Corner Frequency Source Models Used in SMSIM

Generalization of 2-Corner Frequency Source Models Used in SMSIM Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville

More information

APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS

APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS TOPIC 2: MATHEMATICAL LANGUAGE NUMBER AND ALGEBRA You shoul unerstn these mthemtil terms, n e le to use them ppropritely: ² ition, sutrtion, multiplition, ivision ² sum, ifferene, prout, quotient ² inex

More information

Outline Data Structures and Algorithms. Data compression. Data compression. Lossy vs. Lossless. Data Compression

Outline Data Structures and Algorithms. Data compression. Data compression. Lossy vs. Lossless. Data Compression 5-2 Dt Strutures n Algorithms Dt Compression n Huffmn s Algorithm th Fe 2003 Rjshekr Rey Outline Dt ompression Lossy n lossless Exmples Forml view Coes Definition Fixe length vs. vrile length Huffmn s

More information

Learning Partially Observable Markov Models from First Passage Times

Learning Partially Observable Markov Models from First Passage Times Lerning Prtilly Oservle Mrkov s from First Pssge s Jérôme Cllut nd Pierre Dupont Europen Conferene on Mhine Lerning (ECML) 8 Septemer 7 Outline. FPT in models nd sequenes. Prtilly Oservle Mrkov s (POMMs).

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

Generalized Kronecker Product and Its Application

Generalized Kronecker Product and Its Application Vol. 1, No. 1 ISSN: 1916-9795 Generlize Kroneker Prout n Its Applition Xingxing Liu Shool of mthemtis n omputer Siene Ynn University Shnxi 716000, Chin E-mil: lxx6407@163.om Astrt In this pper, we promote

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

CSC2542 State-Space Planning

CSC2542 State-Space Planning CSC2542 Stte-Spe Plnning Sheil MIlrith Deprtment of Computer Siene University of Toronto Fll 2010 1 Aknowlegements Some the slies use in this ourse re moifitions of Dn Nu s leture slies for the textook

More information

Ranking Generalized Fuzzy Numbers using centroid of centroids

Ranking Generalized Fuzzy Numbers using centroid of centroids Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July ning Generlize Fuzzy Numers using entroi of entrois S.Suresh u Y.L.P. Thorni N.vi Shnr Dept. of pplie Mthemtis GIS GITM University Vishptnm

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

Grade 6. Mathematics. Student Booklet SPRING 2008 RELEASED ASSESSMENT QUESTIONS. Assessment of Reading,Writing and Mathematics, Junior Division

Grade 6. Mathematics. Student Booklet SPRING 2008 RELEASED ASSESSMENT QUESTIONS. Assessment of Reading,Writing and Mathematics, Junior Division Gre 6 Assessment of Reing,Writing n Mthemtis, Junior Division Stuent Booklet Mthemtis SPRING 2008 RELEASED ASSESSMENT QUESTIONS Plese note: The formt of these ooklets is slightly ifferent from tht use

More information

New centroid index for ordering fuzzy numbers

New centroid index for ordering fuzzy numbers Interntionl Sientifi Journl Journl of Mmtis http://mmtissientifi-journlom New entroi inex for orering numers Tyee Hjjri Deprtment of Mmtis, Firoozkooh Brnh, Islmi z University, Firoozkooh, Irn Emil: tyeehjjri@yhooom

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

Aperiodic tilings and substitutions

Aperiodic tilings and substitutions Aperioi tilings n sustitutions Niols Ollinger LIFO, Université Orléns Journées SDA2, Amiens June 12th, 2013 The Domino Prolem (DP) Assume we re given finite set of squre pltes of the sme size with eges

More information

Factorising FACTORISING.

Factorising FACTORISING. Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will

More information

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment

More information

Geodesics on Regular Polyhedra with Endpoints at the Vertices

Geodesics on Regular Polyhedra with Endpoints at the Vertices Arnol Mth J (2016) 2:201 211 DOI 101007/s40598-016-0040-z RESEARCH CONTRIBUTION Geoesis on Regulr Polyher with Enpoints t the Verties Dmitry Fuhs 1 To Sergei Thnikov on the osion of his 60th irthy Reeive:

More information

DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS

DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS Krgujev Journl of Mthemtis Volume 38() (204), Pges 35 49. DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS MOHAMMAD W. ALOMARI Abstrt. In this pper, severl bouns for the ifferene between two Riemn-

More information

A Non-parametric Approach in Testing Higher Order Interactions

A Non-parametric Approach in Testing Higher Order Interactions A Non-prmetri Approh in Testing igher Order Intertions G. Bkeerthn Deprtment of Mthemtis, Fulty of Siene Estern University, Chenkldy, Sri Lnk nd S. Smit Deprtment of Crop Siene, Fulty of Agriulture University

More information

APPLICATION OF STRASSEN S ALGORITHM IN RHOTRIX ROW-COLUMN MULTIPLICATION

APPLICATION OF STRASSEN S ALGORITHM IN RHOTRIX ROW-COLUMN MULTIPLICATION Informtion Tehnology for People-entre Development (ITePED ) APPLIATION OF STRASSEN S ALGORITHM IN RHOTRIX ROW-OLUMN MULTIPLIATION Ezugwu E. Aslom 1, Aullhi M., Sni B. 3 n Juniu B. Shlu 4 Deprtment of Mthemtis,

More information

Solving the Class Diagram Restructuring Transformation Case with FunnyQT

Solving the Class Diagram Restructuring Transformation Case with FunnyQT olving the lss Digrm Restruturing Trnsformtion se with FunnyQT Tssilo Horn horn@uni-kolenz.e Institute for oftwre Tehnology, University Kolenz-Lnu, Germny FunnyQT is moel querying n moel trnsformtion lirry

More information

Hyers-Ulam stability of Pielou logistic difference equation

Hyers-Ulam stability of Pielou logistic difference equation vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo

More information

Composite Pattern Matching in Time Series

Composite Pattern Matching in Time Series Composite Pttern Mthing in Time Series Asif Slekin, 1 M. Mustfizur Rhmn, 1 n Rihnul Islm 1 1 Deprtment of Computer Siene n Engineering, Bnglesh University of Engineering n Tehnology Dhk-1000, Bnglesh slekin@gmil.om

More information

Active Contour Model driven by Globally Signed Region Pressure Force

Active Contour Model driven by Globally Signed Region Pressure Force Ative Contour Moel riven y Glolly Signe Region Pressure Fore Mohmme M. Aelsme n Sotirios A. Tsftris IMT Institute for Avne Stuies, Pizz S. Ponzino, 6, 55100 Lu, Itly mohmme.elsme@imtlu.it; s.tsftris@imtlu.it

More information

TOPIC: LINEAR ALGEBRA MATRICES

TOPIC: LINEAR ALGEBRA MATRICES Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

Applied. Grade 9 Assessment of Mathematics. Multiple-Choice Items. Winter 2005

Applied. Grade 9 Assessment of Mathematics. Multiple-Choice Items. Winter 2005 Applie Gre 9 Assessment of Mthemtis Multiple-Choie Items Winter 2005 Plese note: The formt of these ooklets is slightly ifferent from tht use for the ssessment. The items themselves remin the sme. . Multiple-Choie

More information

The DOACROSS statement

The DOACROSS statement The DOACROSS sttement Is prllel loop similr to DOALL, ut it llows prouer-onsumer type of synhroniztion. Synhroniztion is llowe from lower to higher itertions sine it is ssume tht lower itertions re selete

More information

Chapter 8 Roots and Radicals

Chapter 8 Roots and Radicals Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi

More information

Research Article. ISSN (Print) *Corresponding author Askari, A

Research Article. ISSN (Print) *Corresponding author Askari, A Sholrs Journl of Engineering n Tehnology (SJET) Sh. J. Eng. Teh., 4; (6B):84-846 Sholrs Aemi n Sientifi Pulisher (An Interntionl Pulisher for Aemi n Sientifi Resoures) www.sspulisher.om ISSN 3-435X (Online)

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

SEMI-EXCIRCLE OF QUADRILATERAL

SEMI-EXCIRCLE OF QUADRILATERAL JP Journl of Mthemtil Sienes Volume 5, Issue &, 05, Pges - 05 Ishn Pulishing House This pper is ville online t http://wwwiphsiom SEMI-EXCIRCLE OF QUADRILATERAL MASHADI, SRI GEMAWATI, HASRIATI AND HESY

More information

Alpha Algorithm: Limitations

Alpha Algorithm: Limitations Proess Mining: Dt Siene in Ation Alph Algorithm: Limittions prof.dr.ir. Wil vn der Alst www.proessmining.org Let L e n event log over T. α(l) is defined s follows. 1. T L = { t T σ L t σ}, 2. T I = { t

More information

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd

More information

For a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then

For a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:

More information

Chapter 4 State-Space Planning

Chapter 4 State-Space Planning Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different

More information

Automata and Regular Languages

Automata and Regular Languages Chpter 9 Automt n Regulr Lnguges 9. Introution This hpter looks t mthemtil moels of omputtion n lnguges tht esrie them. The moel-lnguge reltionship hs multiple levels. We shll explore the simplest level,

More information

The Stirling Engine: The Heat Engine

The Stirling Engine: The Heat Engine Memoril University of Newfounln Deprtment of Physis n Physil Oenogrphy Physis 2053 Lortory he Stirling Engine: he Het Engine Do not ttempt to operte the engine without supervision. Introution Het engines

More information

Learning Objectives of Module 2 (Algebra and Calculus) Notes:

Learning Objectives of Module 2 (Algebra and Calculus) Notes: 67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under

More information

3D modelling of heating of thermionic cathodes by high-pressure arc plasmas

3D modelling of heating of thermionic cathodes by high-pressure arc plasmas INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 39 (2006) 2124 2134 oi:10.1088/0022-3727/39/10/024 3D moelling of heting of thermioni thoes y high-pressure

More information

Electronic Circuits I Revision after midterm

Electronic Circuits I Revision after midterm Eletroni Ciruits I Revision fter miterm Dr. Ahme ElShfee, ACU : Fll 2018, Eletroni Ciruits I -1 / 14 - MCQ1 # Question If the frequeny of the input voltge in Figure 2 36 is inrese, the output voltge will

More information

VISIBLE AND INFRARED ABSORPTION SPECTRA OF COVERING MATERIALS FOR SOLAR COLLECTORS

VISIBLE AND INFRARED ABSORPTION SPECTRA OF COVERING MATERIALS FOR SOLAR COLLECTORS AGRICULTURAL ENGINEERING VISIBLE AND INFRARED ABSORPTION SPECTRA OF COVERING MATERIALS FOR SOLAR COLLECTORS Ltvi University of Agriulture E-mil: ilze.pelee@llu.lv Astrt Use of solr energy inreses every

More information

Amphenol High Speed Interconnects

Amphenol High Speed Interconnects MTERIL:. EN ESRIPTION TE PPROVE PROPOSL UG 28/1 J.SI GE: OPPER LLO PLTING OPTION =2: 2.5um MIN.NIKEL PLTING OPTION =:.81um MIN. MTTE TIN OVER 1.27um MIN. NIKEL. OTTOM GE GSKET: ONUTIVE RUER GSKET UL-9V-0

More information