Models - Repostores of Knowledge (Proceedngs ModelCARE20 held at Lepzg, Germany, n September 20) (IAHS Publ. 3XX, 20X). Smultaneous estmaton of groundwater recharge rates, assocated zone structures, and hydraulc conductvty values usng fuzzy c-means clusterng and harmony search optmsaton algorthm: A case study of the Tahtal watershed M. TAMER AYVAZ & ALPER ELÇI 2 Department of Cvl Engneerng, Pamukkale Unversty, 20070 Denzl, TURKEY 2 Department of Envronmental Engneerng, Dokuz Eylül Unversty, 3560 Buca-Izmr, TURKEY alper.elc@deu.edu.tr Abstract The am of ths study s to present a lnked smulaton-optmsaton model to estmate the groundwater recharge rates, ther assocated zone structures, and hydraulc conductvty values for regonal, steady-state groundwater flow models. For the zone structure estmaton problem the fuzzy c-means clusterng (FCM) method was used. Accordng to our current knowledge, ths s the frst tme that ths method was mplemented for groundwater recharge zonaton problems. The assocaton of zone structures wth the spatal dstrbuton of groundwater recharge rates was then accomplshed usng an optmsaton approach where the heurstc harmony search (HS) algorthm was used due to ts effcency n fndng global or near global optmum solutons. Snce the soluton was obtaned by a heurstc algorthm, the optmsaton process dd not requre an ntal soluton, whch s an advantage of the proposed approach. The HS based optmsaton model determnes the shape of zone structures, ther correspondng recharge rates and hydraulc conductvty values by mnmzng the root mean square error () between smulated and observed head values at observaton wells and sprngs, respectvely. To determne the best recharge zone structure, the dentfcaton procedure starts wth computaton of one zone and systematcally ncreased the zone number untl the optmum zone structure s dentfed. Subsequently, the performance of the proposed smulatonoptmsaton model was evaluated on the Tahtal) watershed, an urban watershed for whch a seasonal steadystate groundwater flow model was developed for a prevous study. The results of our study demonstrated that the proposed smulaton-optmsaton model s an effectve way to calbrate the groundwater flow models for the cases where tangble nformaton about the groundwater recharge dstrbuton does not exst. Key words groundwater recharge; zone structure estmaton; optmsaton; harmony search INTRODUCTION The spatal dstrbutons of groundwater recharge rates and hydraulc conductvtes are key propertes of groundwater flow models. For occasons when feld data or measurements for these parameters are absent and cannot be obtaned durng the tmeframe gven for the modellng job, numercal estmaton methods can be mplemented. It s the objectve of ths study to propose a procedure to estmate groundwater recharge rates wth the assocated zone structure and hydraulc conductvtes for steady-state groundwater flow models. The proposed procedure nvolves the adaptaton of algorthms that were n the past appled n other felds of scence. Here, the harmony search algorthm s used n combnaton wth the fuzzy c-means clusterng algorthm to determne zone structures and values for hydraulc conductvty and recharge. The applcablty of the entre procedure was demonstrated on the sem-urban Tahtal) watershed n Izmr-Turkey, whch s a key component of Izmr s water supply system. The Tahtal) dam reservor (38 08 N; 27 06 E) s located 40 km south of Izmr and meets about 36% of the cty s total water demand. The watershed of the reservor has an area of 550 km 2 and s a sub-watershed of the larger K. Menderes rver watershed (Fg.). Elç et al. (200) prevously presented results of a seasonal, steady-state groundwater flow model, for whch model parameters were obtaned wth the parameter estmaton code, PEST (Doherty, 2004). Therefore, another objectve of ths study s to compare parameters obtaned wth the proposed procedure to prevously obtaned ones. Copyrght 20X IAHS Press
2 Ayvaz and Elç Fg. General locaton map of applcaton area showng the groundwater flow model boundares and the Tahtal) watershed Fg. 2 Geologcal map of the study area and locatons of the groundwater level montorng wells. Formatons represent at the same tme the hydraulc conductvty structure used n the groundwater model. (K : alluval, K 2 : karstc lmestone, K 3 : flysch, K 4 : tuff, K 5 : conglomerates, K 6 : clayey lmestone) MODEL DEVELOPMENT Optmsaton Model: Harmony Search Algorthm (HS) HS, frst proposed by Geem et al. (200), s a heurstc optmsaton algorthm whch gets ts bass from the muscal processes. It s well known that the purpose of the muscal processes s to seek a muscally pleased harmony through makng several mprovsatons (Yang, 2009). Although HS s a newly proposed optmsaton algorthm, t has been appled to many dfferent problems ncludng water-related applcatons, structural desgn, nformaton technology, transport related problems, thermal and energy problems, and many other applcatons. The state-of-the-art n the structure of HS algorthm and overvew of ts applcatons and developments can be found at Ingram and Zhang (2009) and Geem (200). The mathematcal statement of HS s as follows: Let HMS be the harmony memory sze, N be the number of decson varables, N N = x x= x be the newly generated soluton vector. x { } HMS j be the soluton vectors, and { j} = j= Usng these parameters, soluton of an optmsaton problem s performed based on the followng scheme: HMS. Intalzaton of HM: Generate ntal soluton vectors as many as HMS, x x. 2. Generate a new soluton vector x for each x j : Rnd [, HMS] wth probablty HMCR select xj from memory, x j = xj 3. Ptch adjustment: For each x j : wth probablty PAR change xj as, xj = xj ± bw Rnd(0,). wth probablty (GPAR) do nothng. wth probablty (GHMCR) select a new random value from the possble range. 4. If x s better than the worst x n harmony memory, replace x wth x. 5. Repeat step 2 to 5 untl the gven termnaton crteron s satsfed. j=
Smultaneous dentfcaton of groundwater recharge rates and zone structures usng fuzzy c-means clusterng and harmony search optmzaton algorthm: A case study of the Tahtal! watershed 3 As can be seen from the computatonal scheme gven above, HS requres some soluton parameters whch are Harmony Memory Sze (HMS), Harmony Memory Consderng Rate (HMCR), Ptch Adjustng Rate (PAR), and dstance bandwdth (bw). Note that the HM s a matrx where decson varables and correspondng objectve functon values are stored. The HMCR s the probablty of selectng any harmony from HM. If HMCR s selected too low, only few elte harmones are selected and the algorthm can converge too slowly. On the other hand, f HMCR s selected too hgh, the ptches n HM are mostly used and other possbltes are not explored well (Yang, 2009). If the generated decson varable s selected from the HM, an evaluaton for the requrement of ptch adjustment s necessary. Ths evaluaton s performed usng the PAR parameter whch s the probablty of makng ptch adjustng. Ptch adjustng s a process that s analogous of takng the slghtly neghbour value based on the predefned bandwdth (bw). The ptch adjustng process s smlar to the mutaton operator n genetc algorthm, whch mantans the dversty of populaton (Geem et al., 200). Based on the experence of the authors, HMS=0, HMCR=0.95, and PAR=0.50 are approprate values to solve many optmsaton problems dealng wth groundwater modellng (Ayvaz, 2009; 200). Estmaton of the Groundwater Recharge Zone Structure The recharge zone structure of the model doman s determned usng fuzzy c-means (FCM) clusterng algorthm (Bezdek, 98). In FCM algorthm, fuzzy membershp values are assgned to each data pont whch s related to the relatve dstance of that pont to the cluster centers. FCM provdes a procedure to group the data ponts that populate some multdmensonal space nto a specfc number of dfferent clusters (Ayvaz, 2007). Although the FCM algorthm s extensvely used n many pattern classfcaton and mage processng studes, to our knowledge there s no publshed applcaton example for the groundwater recharge zone structure estmaton problem. The mathematcal statement of FCM whch s modfed for the groundwater recharge zone structure estmaton problem can be summarsed as follows: Let nx and ny be the number of fnte dfference grd ponts of the MODFLOW model n x = X n n x Y = Y y be the vectors that contan the and y drectons, respectvely, { } X and { j} = j= locatons of grd ponts n x and y drectons, respectvely, and c be the number of clusters n whch recharge rates are assumed to be homogeneous (hereafter the term of zone s used nstead of cluster ). Zonaton of the groundwater recharge dstrbuton s performed by usng the 3- n n x y c dmensonal fuzzy partton matrx u = u jk = j= k= such that: 0u jk ; c ujk = ; k= n x n y ujk > 0 () = j= where ujk represents the fuzzy membershp value between the (,j) th grd pont and k th zone ˆ = Xˆ c Yˆ = Yˆ c be the zone centers to be determned by the structure. Let X { k} and { k} k = k= optmsaton model n x and y drectons, respectvely. The elements of the fuzzy partton matrx are updated usng the determned zone centers as follows:
4 Ayvaz and Elç u jk 2 2 mˆ c X ˆ ˆ Xk Yj Y + k = 2 2 t= X ˆ ˆ Xt + Yj Yt (2) where O O s the Eucldean norm, and ˆm s the degree of fuzzfcaton ( m ˆ = 2 ). It should be noted that f the calculated membershp value of a grd pont usng Equaton (2) has a maxmum value, then ths grd pont s assgned to ths zone (Wang and Xue, 2002). By applyng ths procedure to all the fnte dfference grd ponts, the flow doman can be parttoned nto c zones. After ths parttonng process, homogeneous groundwater recharge rates are assgned to each zone by the optmsaton model, and the aqufer s response s determned by performng a MODFLOW run. Problem Formulaton and Search Procedure The purpose of applyng the proposed smulaton-optmsaton procedure to the Tahtal) watershed model s to smultaneously estmate the groundwater recharge zone structure, assocated recharge rates, and unform hydraulc conductvty values wthn the sx geologcal formatons shown n Fg. 2. Ths problem can be formulated as an optmsaton problem n whch HS randomly generates the zone centers; FCM bulds up the zone structures; and fnally, randomly generated recharge rates and hydraulc conductvty values are assgned to the correspondng zone structures. Based on the errors for calculated hydraulc head values, zone centers, assocated recharge rates, and hydraulc conductvty values are modfed by the HS-based optmsaton model. The objectve of the optmsaton model s to mnmse the root mean squared error () between the smulated and observed hydraulc head values at the montorng wells shown n Fgure 2. Ths problem can be mathematcally stated as follows: c ( ) ( ) ( ˆ ˆ c c P Xk Yk) mn = +, (3) k = nw c c n w = ( h ) 2 ( ) = h ( ) % (4)!! h!! h Kh + Kh = W R! x! x! y! y ( ˆ ˆ k, Yk) P X #$ 0 f ( Xˆ, ˆ k Yk) s n actve cell = % $ R f ( ˆ, ˆ &" Xk Yk) s n nactve cell (5) (6) ' 2 '' c (7) where c s the soluton of the problem wth c recharge zones, ( c) s the root mean square error for the soluton of c, * ( c) penalsed objectve functon for the soluton of c, h ( c) s the smulated hydraulc head value at observaton well for the soluton of c, h % s the observed hydraulc head value at observaton well, n w s the number of observaton wells (n w =5), h s the hydraulc head over the flow doman, W s the snks/source term due to pumpng, R s the set of groundwater recharge rates to be estmated such that R( { R, R,, Rc}, K s the set of hydraulc 2
Smultaneous dentfcaton of groundwater recharge rates and zone structures usng fuzzy c-means clusterng and harmony search optmzaton algorthm: A case study of the Tahtal! watershed ( 2 6, ( ˆ ˆ k, k) conductvtes to be estmated such that K { K, K,, K } P X Y s the penalty functon dependng on the locatons of zone centers, 7 s the penalty parameter, and R s the nearest dstance to the model boundary n terms of the row and column numbers of the fnte dfference grd (Fgure 3). As can be seen from Equatons (3) to (7), the groundwater flow process enters the problem n Equaton (5) for unknown R and K dstrbutons. These dstrbutons are determned by the optmsaton model and passed on to MODFLOW to obtan the soluton for groundwater flow n the study area. It should be noted that the reason for usng the penalty functon gven n Equaton (6) s the rregular shape of the modellng doman. All the grd cells outsde the model boundary are specfed as nactve cells, whch are shown as the dark shaded area n Fg. 3. Although these cells appear n the fnte dfference grd structure of the MODFLOW model, they are excluded from the numercal soluton. Therefore, these nactve cells must be also excluded from the search space of the zone centers. Equaton (6) states that f a zone centre s located n an nactve cell, the calculated objectve functon value s penalzed wth 7R. The values of R ncrease as the zone centers move away from the model boundares. The value of 7 s mostly arbtrary and problem dependent. Our trals show that 7=00 can be used for the mplementaton of the penalty functon gven n Equaton (6). The decson varables of the optmsaton model are the locatons of the zone centers, assocated recharge values for each zone, and unform hydraulc conductvty values for each of the sx pre-defned geologcal zones. 5 Model Boundary ) Values Inactve Cells Fg. 3 The model doman and R values used for the penalty functon gven n Equaton (6) Although the proposed smulaton-optmsaton model may solve the problem based on the soluton scheme gven n Equaton (3) to (7), ths mathematcal formulaton s vald only for cases where the number of zones (c) s known a pror. However, recharge zone structures, ther numbers, and the assocated recharge rates are unknown for the most cases. Therefore, t s necessary to determne the number of zones such that the eventually dentfed zone structure optmally represents the feld data. Wth ths purpose, the zone structure estmaton problem starts wth one zone, and then, systematcally ncreases the zone number untl the best soluton s obtaned. Furthermore, each successve soluton for dfferent zone numbers requres three addtonal decson varables (one s for recharge rate and two for zone centre coordnates). However, when the number of decson varables ncreases, there s a greater chance of producng local optmum
6 Ayvaz and Elç solutons due to the ncreased dmenson of soluton space (Huang and Mayer, 997). Ths stuaton also leads to less relable solutons. For such cases, the fnal value of the objectve functon may ncrease although '0, whle c'u (Ayvaz, 2007). Therefore, fnal dentfed parameter values, zone structures, and objectve functon values are evaluated altogether to decde whch successve zone structure best represents feld condtons. Our tral runs ndcate that the value of the objectve functon does not mprove sgnfcantly after about 5,000 teratons of HS. Therefore, the maxmum number of teratons s set to 20,000. Completng 20,000 teratons of HS takes about 2 hours on a workstaton wth Intel Xeon 3.07 GHz processor and 6 GB RAM. IDENTIFICATION RESULTS Fg. 4 shows the dentfed groundwater recharge zone structures for the solutons of 2 to 6 and the recharge zone structure orgnally used by Elç et al. (200). As can be seen from Fgure 4, centres of the dentfed zones reman nsde the flow doman by vrtue of the penalty functon mplementaton. Ths result also mples that the fnal objectve functon values do not nclude any penalty term (.e. * =). (a) (b) (c) (d) (e) (f) Fg. 4 Comparson of the dentfed zone structures; (a) to (e): for to 2 6 (small crcles correspond to zone centers); (f): The zone structure used by Elç et al. (200) (shaded area represents the Tahtal) watershed) Summary of the dentfed hydraulc conductvty values, recharge rates, and fnal * values for solutons to 6, and the results by Elç et al. (200) are gven n Table. The smulatonoptmsaton model calculates hydraulc conductvty values that are comparable between all solutons. On the other hand, the dentfed recharge rates are all dfferent because the zone structures for recharge evolve durng the optmsaton, whle the zone structure for hydraulc conductvty s fxed. Regardng fnal * values after 20,000 teratons, t can be observed that the largest * value (6.8) s obtaned for where t s assumed that the flow doman takes a
Smultaneous dentfcaton of groundwater recharge rates and zone structures usng fuzzy c-means clusterng and harmony search optmzaton algorthm: A case study of the Tahtal! watershed 7 unform recharge wth a rate of 2.65 0-4 m/d. For ths soluton, the number of decson varables s seven, sx for conductvtes and one for unform recharge rate. After ths soluton, the value of * decreases as the soluton approaches 4, and ncreases agan for 5 and 6. As mentoned earler, theoretcally the ncrease n the zone numbers should result n the decrease n the correspondng * values. For such cases, however, the relablty of the dentwed parameters generally decreases snce more unknown parameters need to be determned (Ayvaz, 2007). Therefore, by consderng the dentfed parameter values, zone structures, and the fnal * values, t can be concluded that the relablty of the dentfed solutons after 4 tends to decrease. Thus, the four-zone structure ( 4) s selected as the best zone structure for the estmaton problem dscussed here (Fgure 4(c)). Table Summary of the dentfed hydraulc conductvty values, recharge rates and fnal values Identfed Solutons Elç et al. Parameters 2 3 4 5 6 (200) K 20.8 22.2 20.6 25.87 26.35 24.38 7.06 K 2 0.0 0.0 0.0 0.0 0.0 0.28 0.0 K 3.00.00 0.54 0.9 0.95 0.65 0.30 K 4 5.74 6.07 7.49 7.57 6.99 6.46 7.09 K 5 5.00 5.00 5.00 4.98 4.38 4.47.9 K 6 3.58 3.62 2.6 2.56.64 3.06.35 Hydraulc Conductvtes (m/d) Groundwater Recharge Rates (m/d) R 2.65E-04.05E-03.00E-0 9.53E-04 4.02E-05 3.23E-06 6.27E-05 b R 2-2.66E-04.38E-04 2.3E-05.00E-0 6.56E-04.27E-04 b R 3 - - 6.2E-04 9.05E-04 5.78E-04 6.97E-04 5.00E-04 b R 4 - - -.33E-04 6.3E-04.3E-04 9.02E-05 c R 5 - - - - 8.70E-04.03E-03 - R 6 - - - - - 7.55E-05 - Fnal * value 6.8 5.97 2.96.90 2.55 3.30 6.40 a a Ths value equals to and does not nclude the penalty functon n Equaton (7) b These values were calculated usng the PEST model for fxed recharge and conductvty zone structures c Ths value was calculated based on a external transent precptaton-runoff model In the prevous modellng study by Elç et al. (200), the calbraton of the same groundwater flow model was performed by adjustng the recharge rates and hydraulc conductvty values usng the PEST parameter estmaton code, whle keepng the recharge rate for the zone representng the Tahtal) watershed constant at a value that was obtaned by an ndependent precptaton-runoff model. For that study the hydraulc conductvty zone structure was based on the geology of the study area and the four-zone recharge zone structure was manually created based on land use/land cover and lthology nformaton. Comparson of results obtaned by Elç et al. (200) wth the results for 4 shows that optmsed hydraulc conductvty values n ths study are n the same order of magntude, except for zone 2 (K 2 ). However, ths s not the case for recharge rates, as they are dfferent for both studes. Ths dfference can be explaned by the dfferent outcome of zone structures n both studes. It had to be assumed by Elç et al., that the recharge rate for the entre watershed (zone 4 n Fgure 4 (f)) s unform snce the precptaton-runoff model was a lumped model. In the current model, however, ths part of the model doman was splt nto other zones, each allowed to have dfferent recharge rates. Elç et al. (200) obtaned a fnal value as 6.40 for the four-zone structure gven n Fgure 4(f), whch ndcates a less optmsed soluton compared to the * value (.90) of the 4 soluton gven n Fgure 4(c). As can be seen from these results, the fnal * value decreases by 27% through the use of the smulaton-optmsaton procedure when compared to Elç et al. (200). Based on the error evaluaton, t can be concluded that the groundwater flow model s mproved wth the proposed procedure. CONCLUSIONS A coupled smulaton-optmsaton model s developed for the smultaneous estmaton of groundwater recharge zone structure, ther assocated recharge rates and hydraulc conductvty
8 Ayvaz and Elç values. In ths model, MODFLOW s used to perform the steady-state groundwater flow calculatons. The assocaton of recharge zone structures wth the recharge rates s accomplshed by lnkng the FCM clusterng algorthm wth the MODFLOW n the smulaton model. Ths model s then ntegrated to an optmsaton model where heurstc HS algorthm s used. The man objectve of the HS based optmsaton model s to mnmze the root mean square error whch s calculated between the smulated and observed hydraulc head values at avalable observaton wells by adjustng the zone centres, assocated recharge rates wthn each generated zone, and unform hydraulc conductvty values. The applcablty of the developed model s evaluated n a case study for the Tahtal) watershed (Izmr-Turkey) and the estmaton results are compared to prevous modellng results for the same model doman that were obtaned wth a dfferent optmsaton approach. Comparson of the results ndcate that the developed model may be an effectve way n calbratng steady-state groundwater flow models, where tangble nformaton about the groundwater recharge dstrbuton does not exst. Acknowledgements Ths study s based on the work supported by The Turksh Academy of Scences (TÜBA) - The Young Scentsts Award Programme (GEBIP). The frst author would lke to thank TÜBA for ther support of ths study. REFERENCES Ayvaz, M. T. (2007) Smultaneous determnaton of aqufer parameters and zone structures wth fuzzy c-means clusterng and meta-heurstc harmony search algorthm. Adv. Water Resour. 30, 2326-2338. Ayvaz, M. T. (2009) Applcaton of harmony search algorthm to the soluton of groundwater management models. Adv. Water Resour. 32(6), 96-924. Ayvaz, M. T. (200) A lnked smulaton-optmzaton model for solvng the unknown groundwater polluton source dentfcaton problems. J. Cont. Hydrol. 7(-4), 46-59. Bezdek, J. C. (98) Pattern recognton wth fuzzy objectve functon algorthms. Advanced applcatons n pattern recognton. Plenum Press, New York, USA. Doherty, J. (2004) PEST Model ndependent parameter estmaton. User Manual. Watermark Numercal Computng. Elç, A. Karada[, D. & F)st)ko]lu, O. (200) The combned use of MODFLOW and precptaton-runoff modelng to smulate groundwater flow n a dffuse-polluton prone watershed. Water Sc. Tech. 62(), 80-88. Geem, Z. W., Km, J. H. & Loganathan, G. V. (200) A new heurstc optmzaton algorthm: harmony search. Smulaton. 76(2), 60 68. Geem, Z.W. (200) State-of-the-art n the structure of harmony search algorthm. In: Recent Advances n Harmony Search Algorthm (ed. By Z. W. Geem), Sprnger. Huang, C. & Mayer, A. S. (997) Pump-and-treat optmzaton usng well locatons and pumpng rates as decson varables. Water Resour. Res. 33(5), 00-02. Ingram, G. & Zhang, T. (2009) Overvew of applcatons and developments n the harmony search algorthm. In: Musc- Inspred Harmony Search Algorthm: Theory and Applcatons (ed. By Z. W. Geem), Sprnger. Wang, H. & Xue, D. (2002) An ntellgent zone-based delvery schedulng approach. Comp. Industry 48(2), 09-25. Yang, X. S. (2009) Harmony search as a metaheurstc algorthm. In: Musc-Inspred Harmony Search Algorthm: Theory and Applcatons (ed. By Z. W. Geem), Sprnger.