Dissolved Oxygen in Streams: Parameter estimation for the Delta method

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1 Dissolved Oxygen in Strems: Prmeter estimtion or the Delt method Rjesh Srivstv Associte Proessor, Deprtment o Civil Engineering, Indin Institute o Technology Knpur, Knpur 0806, Indi. Abstrct The Delt method is one o the techniques used or estimting the reertion coeicient, respirtion rte, nd production rte in strems rom the diurnl dissolved oxygen (DO) proile. Recently, n pproximte Delt method hs been proposed which llows or pproximte, but simpler, computtion o these prmeters. We suggest some modiictions in the pproximte Delt method to improve its ccurcy. It is shown tht the proposed method is more ccurte or wide rnge o prmeter sets. Introduction Dissolved oxygen (DO) in strem is useul indictor o its wter qulity. Three min prmeters which ect the diurnl DO vrition re the reertion coeicient, production rte, nd respirtion rte. Estimtion o these prmeters rom mesured DO proile hs been the subject o mny studies (e.g., Odum 956, O Connor nd Di Toro 970, Hornberger nd Kelly 97, Kelly et l. 974, Hornberger nd Kelly 975, Chpr nd Di Toro 99). A recent nd excellent review o vrious modelling techniques is provided by Cox (003). Assuming negligible longitudinl oxygen grdients, constnt strem temperture, nd primry production rte o the orm o hl-sinusoid within the photoperiod, Chpr nd Di Toro (99) obtined periodic nlyticl solution or the diurnl dissolved oxygen (DO) proile. The method ws clled the delt method, since n importnt prmeter used in the nlysis ws the diurnl rnge o vrition o the dissolved oxygen, denoted by. The method, though hving its limittions (Chpr nd Di Toro 99, Cohen 99, Erdmnn 99), hs been widely used or nlysing the DO proiles (e.g., Wilcock et l. 995, Gusch et l. 998, Wilcock et l. 998, Jrvie et l. 003, Wng et l. 003, Kim nd Je 004). The orm o the nlyticl solution, while llowing direct computtion o the DO proile or given prmeters (the strem reertion coeicient, respirtion rte, nd verge plnt primry production rte), does not oer explicit closed orm expressions or estimting the prmeter vlues rom the mesured DO proile t sttion. Chpr nd Di Toro (99) thereore used numericl techniques to obtin the prmeters rom chrcteristics o the mesured DO proile nd presented the results in grphicl orm. McBride (00), noting tht the shpe o the grph relting the reertion coeicient to the time lg between the DO mximum nd solr noon is similr to tht o logistic curve, obtined n pproximte eqution or the sme. This work ws lter extended by McBride nd Chpr (005) who proposed nother pproximte eqution (gin, bsed on the logistic curve) relting the dily verge production rte to the diurnl vrition rnge nd the reertion coeicient. This pproximte delt method (ADM) ws pplied to number o ield mesurements nd ws ound to be resonbly ccurte. However, the pproximte equtions were seen to hve lrge errors or some combintion o vlues. For exmple, the pproximte eqution or reertion coeicient my overpredict the vlue by s much s 0% or time lg vlues o bout to hours nd lso results in lrge errors when the lg is close to its theoreticl mximum vlue. Similrly, the pproximte eqution or the production rte is not very ccurte or ertion coeicient vlues lrger thn bout 0 d nd lso or photoperiod length o 8 h. This ws

2 stted to be mthemticl curve itting issue nd better its o simple orm could not be obtined. McBride nd Chpr (005) lso note tht the ADM could predict negtive DO vlues or some combintion o prmeter vlues nd speciied some constrints on the prmeter vlues which would ensure the voidnce o such problems. In this pper we nlyse the ccurcy o the pproximte delt method nd ttempt to improve the ccurcy by choosing little more complex orm o the pproximtion bsed on n nlysis o the nlyticl expression. We lso look closely t the issue o negtive DO predictions nd suggest n lterntive wy o computing the production rte. Mthemticl Bckground The eqution governing the vrition o DO in strem is dc = k ( C st C ) + P R () dt where, C = DO (mg/l); t = time (h); = irst-order strem reertion coeicient (h ); C st = sturted DO (mg/l); P = plnt primry production rte (mg/l/h); nd R = respirtion rte (mg/l/h). Chpr nd Di Toro (99) worked with the DO deicit, D (mg/l), deined s D= Cst C () nd ssumed tht the strem temperture is constnt nd the production rte is given by hl-sinusoid πt πt P = Pv sin 0 t (3) = 0 t T where T is the period length (=4 h); is the photoperiod length (h); P v is the dily-verge production rte (mg/l/h); nd the time, t, is mesured rom sunrise. The nlyticl solution or the DO ws obtined s (written in slightly dierent orm here since we use the DO rther thn the DO deicit in this pper) R πt πt kt C = Cst + PT v σ sin cos + γe 0 t π (4) R ( t ) = Cst + PT v σ + γe e t T k where the dimensionless prmeter groups, which re unctions o nd, re deined s ( T ) + e γ =, σ kt = (5) e + π From the nlyticl solution, it my be demonstrted tht both the minimum nd the mximum o the DO occur within the photoperiod nd the time lg between the DO mximum nd solr noon, φ (see Fig. ), could be obtined by solving (McBride 00) πφ πφ ( φ+ ) π cos sin γe = 0: φ > 0 (6)

3 8 t mx DO, C (mg/l) D Photoperiod, Solr Noon D C mx C C min 0 t min Time since sunrise, t (h) Figure. Deinition sketch o DO proile showing the key prmeters From Eq. (6), it is seen tht the reertion coeicient,, is only unction o φ nd. Chpr nd Di Toro (99) numericlly solved Eq. (6) nd provided grph which could be used or obtining the vlue o or given φ or ew selected vlues o (Fig. ). Wldon (985) proposed pek lg method nd modiied orm (Wldon 99) or estimtion o s 0.68 = 4 ( ) φ ( ) (7) tn( 0.68φ ) where,, rom now onwrds, is in the commonly used units o d. The ccurcy o Eq. (7) ws, however, ound to be uncceptble or some prmeter vlues (Chpr nd Di Toro 99). McBride (00) itted logistic curve to the plot o versus φ or =4 h, nd used photoperiod correction ctor or other vlues o to obtin the ollowing pproximtion: 5.3η φ = 7.5 ηφ where η is the photoperiod correction ctor given by η = (9) 4 It ws ound tht the pproximtion, Eq. (8), works well except or smll nd lrge. McBride nd Chpr (005) extended this concept nd proposed n pproximte delt method which provided n dditionl logistic curve it between the rtio o the diurnl DO = ( C C ), to P v nd the reertion coeicient,, s rnge, mx min 6 = P η + v ( 33 ) 0.85 (8) (0)

4 with P v expressed in the more convenient units o mg/l/d. Thus once is known rom Eq. (8), the verge production rte could be obtined using Eq. (0). However, the pproximtion involved errors s lrge s 5% or vlues rom 0. to d nd much higher or lrger thn 0 d. The respirtion rte, in mg/l/d, ws then obtined rom R = P + k C C () v st ( ) where C is the diurnl verge DO (mg/l). Wng et l. (003) stted tht the delt method could propgte computtionl errors since computed prmeter, which itsel my be subjected to lrge errors, is subsequently used in the estimtion o other prmeters. For exmple, estimted rom Eq. (8) is used to obtin P v through Eq. (0) nd both these re used to estimte R through Eq. (). They proposed n extreme vlue method bsed on the ct tht t the points o extremes o DO (or DO deicit), the slope o the DO curve is zero nd, rom Eq. (), R= P+ k C C or t=t or t () ( st ) min mx Further, ssuming tht the minimum DO occurs during night when P=0 (note tht this ssumption is in contrdiction o the delt method solution which results in the minimum DO little ter sunrise), they obtined, or the respirtion rte, R= k C C (3) nd, or the production rte, ( ) st min ( ) R Cst Cmx Pv = (4) πt πφ cos Comprison with the delt method showed tht the extreme vlue method produced similr results. McBride nd Chpr (005) mention tht the nlyticl solution o Chpr nd Di Toro (99) does not put lower bound on the DO nd it is possible, or certin combintion o prmeter vlues, to get negtive DO vlues. They perormed numericl experiments to suggest the ollowing constrints to void these non-physicl negtive DO vlues: Anlysis 5 R < 40e k > 6d ( k ) P v k R < 4.8e k 6d Fig. shows the percent error in prediction o using Eq. (8) nd provided the motivtion to rrive t better it which should be more ccurte nd my be o little more complicted orm thn Eq. (8). The itting philosophy used ws tht o mtching the symptotic behviour nd then tweking some prmeters to obtin generl improvement in the it. Anlysing Eq. (6), the ollowing limiting cses were obtined (note tht φ nd re in h nd in d ): 0: φ cos π πt (6) 4 : φ k (5)

5 Percent Error (h) Figure. Percent Error in the prediction o. Thick line Eq. (8), Thin line Eq. (8). Numbers ner the curve represent the vlue o (in hours). The orm o Eq. (6) or smll suggested possible itting unction o the orm πφ π cos 4 T = (7) πφ φ sin T which stisies both the symptotic conditions listed in Eq. (6). Dt generted using vlues o equl to 8, 0,, 4, nd 6 h nd rnge o rom 0. to 00 d ws used to estimte the error in the pproximte expression. Use o Eq. (5) produced mximum error o bout % nd it ws, thereore, modiied s π πφ cos = 4 (8) πφ 0.83 πφ sin φ + sin Fig. 3 shows the plot o obtined rom Eq. (8) long with the nlyticl solution nd Fig. shows comprison o the errors obtined rom Eqs. (8) nd (8). As seen rom Fig. 3, or vlues o less thn bout d, smll dierence in φ my cuse dierence o bout n order o mgnitude in (Chpr nd Di Toro 99). Since the mesurement o φ involves some uncertinties, it would probbly not be prudent to come up with n pproximtion which is extremely ccurte or smll vlues o. Thereore, in ll the pproximte expressions derived in this pper, we try to minimize the mximum error or the rnge 00 nd hve resonble error or smller vlues. It is seen tht the perormnce o Eq. (8) is generlly better with the mximum error or lrger vlues being less thn 4%. The dditionl complexity o Eq. (8) my be drwbck rom the point o view o hnd computtions but should not be ctor in computer pplictions.

6 00 0 (d - ) (h) Figure 3. Comprison o the predicted nd theoreticl vlues. Solid line Delt solution, Broken line Eq. (8). Numbers ner the curve re vlues o. Similrly, or the primry production rte, the ollowing limiting vlues were obtined: where, P v 0: cos Pv π π π (9) : π Pv is in d, is in d nd is in h. Chpr nd Di Toro (99) nd Chpr nd McBride (005) expressed s unction o nd k P. However, Wng et l. (003) v pointed tht use o the derived prmeter, which is lredy pproximte, my led to ccumultion o error in the prediction o P v. We eel tht Eq. (0) ws itted bsed on the pproximte nd not the exct vlue nd thereore would not led to propgtion o error. In order to void mbiguity, however, we express s unction o nd φ. Using the sme philosophy s used or derivtion o Eq. (8), we obtin the ollowing pproximtion: πφ φ φ φ sin π φm φm = (0) P 4.85 v πφ π cos sin( 0.55φ ) in which φ m is the mximum vlue o φ or ny given, s obtined rom the smll limit in Eq. (6). Fig. 4 shows comprison o vlues obtined rom Eq. (0) nd those rom the nlyticl solution nd Fig. 5 compres the errors obtined rom Eq. (0) nd tht rom Eq. (0). The mximum error in the use o Eq. (0) is bout 3%. The other lterntive, Eq. (4), ws not considered becuse the ssumption mde by Wng et l. (003) o minimum DO occurring during night time is not consistent with the nlyticl solution nd my led to lrge errors (s shown in the next section) i the minimum DO occurs ter sunrise (the time P v

7 o minimum DO my be s lte s hours ter sunrise or long photoperiod nd smll reertion coeicient). 0.8 D/P v (d) (d - ) Figure 4. Comprison o the predicted nd theoreticl vlues o Pv.Thick line Delt solution, Thin line Eq. (0). Numbers ner the curve re vlues o Percent Error (d - ) Figure 5. Percent error in the predicted vlue o Pv (0). Numbers ner the curve re vlues o..thick line Eq. (0), Thin line Eq. During the derivtion o the pproximte expressions or the production rte, the possibility o hving negtive DO ws not considered. Eq. (5) provides criterion or predicting whether negtive DO will occur. However, rom n ppliction point o view, it my not be s importnt to predict the conditions under which negtive DO occurs s to suggest method to estimte the prmeters under these conditions. The observed DO proile in such cses will show zero vlue or some length o time. We my still ssume the model to be vlid with the modiiction tht the negtive vlues o DO predicted rom the model re tken s zero. However, the diurnl DO rnge,, would now be equl to C mx nd Eqs. (0) nd (0) would underpredict P v. Eq. (4) would be pplicble but, s discussed erlier, the

8 ssumption mde in Eq. (3), o minimum DO occurring beore sunrise, my led to lrge errors. Mking the ssumption tht the verge DO would not be signiicntly ected by the presence o the zero DO segment, we propose below modiied delt method which uses the dierence between the mximum nd verge DO s its prmeter. Using the extreme vlue method o Wng et l. (003) nd the time verged eqution (), it cn be shown tht Pv = () πt πφ cos where = Cmx C (see Fig. ). Fig. 6 shows the theoreticl grph relting to nd φ. Using the pproximte eqution or, Eq. (8), nd djusting the coeicients to obtin minimum error or >, we get πφ φ sin = () Pv 0.7 πφ sin Fig. 6 shows the vlues predicted rom Eq. () nd Fig. 7 shows the reltive error o prediction. The mximum error or lrger reertion coeicients is less thn 4%. We believe tht Eq. (), which we cll the modiied delt method (MDM), would provide better estimte o the production rte since the verge DO would not be signiicntly ected by the theoreticlly negtive DO vlues nd would be subjected to smller errors compred to. In the next section, we pply the modiied delt method to synthetic nd ield dt sets nd show its improved ccurcy. P v D/P v (d) (d - ) Figure 6. Comprison o the predicted nd theoreticl vlues o Pv.Thick line Delt solution, Thin line Eq. (). Numbers ner the curve re vlues o.

9 0 Percent Error (h) Figure 7. Percent error in the vlue o Applictions Pv predicted rom Eq. (). Numbers ner the curve re vlues o. Five cses hve been considered or ppliction o the modiied delt method. Three o these re syntheticlly generted dt nd the other two re bsed on ield mesurements o diurnl DO proiles. Tble presents summry o the key etures o these cses. Note tht the prmeters listed or Mngorong re slightly dierent rom those reported in McBride nd Chpr (005), becuse we obtined these prmeters independently by using the etures o the DO proile listed in Wilcock et l. (998). Tble. Prmeters used in vrious pplictions Dt Set φ C mx C min C C st P v R (h) (h) (mg/l) (mg/l) (mg/l) (mg/l) (d - ) Synthetic I Grnd River Mngorong Strem Synthetic II Synthetic III Synthetic dt set I This ppliction uses the DO proile generted rom the nlyticl solution (Eq. 4) nd compres the reltive ccurcy o prmeters estimted rom dierent methods. Tble lists the results nd shows the improved ccurcy o the proposed method. It is seen tht the modiied delt method, Eq. (), results in slightly lrger error thn tht rom Eq. (0) but both these pproximtions re signiicntly more ccurte thn the pproximte delt method. Fig. 8 shows DO proiles rom the delt method, pproximte delt method, nd the proposed method. Slight improvement is observed when the prmeters obtined rom the proposed method re used.

10 Tble. Results or the dt set Synthetic-I Method (d - ) P v R Percent Error in P v R Delt ADM (Eqs. 8, 0, & ) Proposed (Eqs. 8, 0, & ) MDM (Eqs. 8,, & ) DO (mg/l) Time since sunrise, t (h) Figure 8. Diurnl DO proile or the Synthetic-I dt set. Symbols Delt method, Solid line Proposed method, Dshed line ADM. Grnd River The DO proile observed t Grnd River, Michign, hve been nlyzed in some erlier studies (Chpr nd Di Toro 99, McBride nd Chpr 005). Tble 3 lists the results obtined using dierent schemes o prmeter estimtion. In this cse, the modiied delt method is little more ccurte thn Eq. (0) nd both o them re slightly better thn the ADM. For this, nd the next, ppliction, it should be noted tht the vlue o is not one o the vlues used or generting the pproximte expressions (i.e., 8, 0,, 4, nd 6 h). The results, thereore, provide some conidence tht the proposed pproximtions would work or other vlues o lso.

11 Tble 3. Results or the Grnd River dt Method (d - ) P v R Percent Error in P v R Delt ADM (Eqs. 8, 0, & ) Proposed (Eqs. 8, 0, & ) MDM (Eqs. 8,, & ) Mngorong Strem The DO proile observed t Mngorong strem, New Zelnd, hve been nlyzed in some erlier studies (Wilcock et l. 998, McBride nd Chpr 005). Tble 4 lists the results obtined using dierent schemes o prmeter estimtion. The modiied delt method hs slightly more error thn Eq. (0) but both o these re, gin, more ccurte thn the ADM. Tble 4. Results or the Mngorong Strem Method (d - ) P v R Percent Error in P v R Delt ADM (Eqs. 8, 0, & ) Proposed (Eqs. 8, 0, & ) MDM (Eqs. 8,, & ) Synthetic dt set II This ppliction uses the DO proile generted rom the nlyticl solution (Eq. 4) with the prmeters chosen in such wy s to give negtive DO or some length o time. This negtive DO ws discrded nd the proile ws seen to hve more thn 7 hours o zero DO (Fig. 9). As discussed erlier, this will ect both the minimum nd the verge DO but it is expected tht the verge DO would not be s signiicntly ected s the minimum DO. Tble 5 lists the results nd shows the improved ccurcy o the modiied delt method. Note tht the modiied delt method lso hs lrge error (bout 9%) which is due to the trunction o the DO proile. However, it outperorms the other methods (Fig. 9), especilly in the estimtion o the verge production rte. Using the extreme vlue method o Wng et l. (003), Eqs. (3) nd (4), with the exct vlue o 3 d, we obtin R= 7.00 mg/l/d nd P v =.9 mg/l/d, which hve errors o the order o 30%.

12 Tble 5. Results or the dt set Synthetic-II Method (d - ) P v R Percent Error in P v R Delt ADM (Eqs. 8, 0, & ) Proposed (Eqs. 8, 0, & ) MDM (Eqs. 8,, & ) DO (mg/l) Time since sunrise, t (h) Figure 9. Diurnl DO proile or the Synthetic-II dt set. Symbols Delt method, Solid line Modiied Delt method, Dshed line ADM. Synthetic dt set III As is cler rom Fig., or some combintion o prmeter vlues, the ADM would be more ccurte thn the proposed method. Here we present such cse. Tble 6 lists the results nd shows the better ccurcy o the pproximte delt method. However, the modiied delt method still hs cceptble errors. Using the extreme vlue method o Wng et l. (003), Eqs. (3) nd (4), with the exct vlue o 0.4 d, we obtin R=.93 mg/l/d nd P v =3.54 mg/l/d, which hve errors o the order o 80%.

13 Tble 6. Results or the dt set Synthetic-III Method (d - ) P v R Percent Error in P v R Delt ADM (Eqs. 8, 0, & ) Proposed (Eqs. 8, 0, & ) MDM (Eqs. 8,, & ) Summry nd Conclusions New pproximtions hve been proposed or estimtion o prmeters using the delt method. Although o little more complicted orm thn the previously proposed pproximte delt method, these expressions led to signiicntly improved ccurcy in the estimted vlue o the prmeters. A modiied delt method hs been proposed or cses where the combintion o prmeter vlues would result in negtive DO rom the nlyticl solution. For these cses, insted o using the diurnl rnge o DO vrition, it is proposed tht the devition o the mximum DO rom the men be used in the estimtion o production rte. Vrious dt sets, some synthetic nd some rom the ield observtions, hve been used to show the improvement in ccurcy o prmeter estimtion. Reerences Chpr, S.C., nd Di Toro, D.M. (99). Delt Method or Estimting Primry Production, Respirtion, nd Reertion in Strems. J. Environ. Eng., 7(5), Chpr, S.C., nd Di Toro, D.M. (99). Closure to discussions on Delt Method or Estimting Primry Production, Respirtion, nd Reertion in Strems. J. Environ. Eng., 8(6), Chpr, S.C., nd McBride, G.B. (005). Rpid clcultion o oxygen in strems: Approximte delt method. J. Environ. Eng., 3(3), Cohen, R. H. (99). Discussion o Delt Method or Estimting Primry Production, Respirtion, nd Reertion in Strems. J. Environ. Eng., 8(6), Cox, B. A. (003). A review o dissolved oxygen modelling techniques or lowlnd rivers. The Science o The Totl Environment, 34, Erdmnn, J.B. (99). Discussion o Delt Method or Estimting Primry Production, Respirtion, nd Reertion in Strems. J. Environ. Eng., 8(6), Gusch, H., Armengol, J., Mrti, E., nd Sbter, S. (998). Diurnl vrition in dissolved oxygen nd crbon dioxide in two low-order strems. Wter Reserch, 3 (4), Hornberger, G.M., nd Kelly, M.G. (97). The determintion o primry production in strem using n exct solution to the oxygen blnce eqution. Wter Resour. Bull., 8(4), Hornberger, G.M., nd Kelly, M.G. (975). Atmospheric reertion in river using productivity nlysis. J. Environ. Eng., 0(5),

14 Jrvie, H.P., Love, A.J., Willims, R.J., nd Nel, C. (003). Mesuring in-strem productivity: the potentil o continuous chlorophyll nd dissolved oxygen monitoring or ssessing the ecologicl sttus o surce wters. Wter Sci. Tech., 48(0), Kelly, M.G., Hornberger, G.M., nd Cosby, B.J. (974). Continuous utomted mesurements o rtes o photosynthesis nd respirtion in n undisturbed river community. Limnol. Ocenogr., 9, Kim, K.-S., nd Je, C.-H. (004). Phytoplnkton estimtion using plnt primry production rtes in strems. Env. Geol., 46(), 0-4. McBride, G.B. (00). Clculting strem reertion coeicients rom oxygen proiles. J. Environ. Eng., 8(4), O'Connor, D.J., nd Di Toro, D.M. (970). Photosynthesis nd oxygen blnce in strems. J. Snit. Engrg., 96(), Odum, H.T. (956). Primry production in lowing wters. Limnol. Ocenog., (), Wldon, M.G. (985). Reertion rte estimtion using the lg in dissolved oxygen concentrtion. J. Envir. Sci. Helth, A0(6), Wldon, M.G. (99). Discussion o Delt Method or Estimting Primry Production, Respirtion, nd Reertion in Strems. J. Environ. Eng., 8(6), Wng, H., Hondzo, M., Xu, C., Poole, V., nd Spcie, A. (003). Dissolved oxygen dynmics o strems drining n urbnized nd n griculturl ctchment. Ecol. Model., 60,45-6. Wilcock, R.J., McBride, G.B., Ngels, J.W., nd Northcott, G.L. (995). Wter qulity in polluted lowlnd strem with chroniclly depressed dissolved oxygen: Cuses nd eects. New Zelnd J. Mrine Freshwter Res., 9(), Wilcock, R.J., Ngels, J.W., McBride, G.B., Collier, K.J., Wilson, B.T., nd Huser, B.A. (998). Chrcteristion o lowlnd strems using single-sttion diurnl curve nlysis model with continuous monitoring dt or dissolved oxygen nd temperture. New Zelnd J. Mrine Freshwter Res., 3 (),

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite