SIMGRO Theory and model implementation. Alterrareport Alterra, Green World Research, Wageningen

 Helena Anthony
 10 months ago
 Views:
Transcription
1 SIMGRO Theory and model mlementaton P.E.V. van Walsum A.A. Veldhuzen P. Groenendjk Alterrareort 93. Alterra, Green World Research, Wagenngen
2 ABSTRACT P.E.V. van Walsum, A.A. Veldhuzen, P. Groenendjk SIMGRO 7.2., Theory and model mlementaton. Wagenngen, Alterra. AlterraReort SIMGRO s a modellng framework that can facltate the nvestgaton of varous knds of regonal water management roblems. The framework connects to dverse hydrologc model codes lke MODFLOW for groundwater and SOBEKCF for surface water smulaton. A number of nhouse codes make SIMGRO esecally sutable for modellng stuatons wth shallow groundwater levels n relatvely flat areas, lke n delta regons. The metamodel MetaSWAP s based on a quas steadystate soluton of the Rchards equaton. Its mlementaton can nclude cro growth smulaton wth WOFOST, wth feedback to the hydrologc arameters lke root zone deth and leaf area ndex. The SIMGRO ackage also ncludes a smlfed code for the smulaton of surface water rocesses. A secal feature of the framework s the oton for model coulng va a shared state varable that s alternately udated by the connected models. Keywords: ntegrated water management, mechanstc, dstrbuted, dynamc, saturatedunsaturated flow, surface water, dranage; semmlct ISSN Alterra, P.O. Box 47, NL6700 AA Wagenngen (The Netherlands). Phone: ; fax: ; emal: No art of ths ublcaton may be reroduced or ublshed n any form or by any means, or stored n a data base or retreval system, wthout the wrtten ermsson of Alterra. Alterra assumes no lablty for any losses resultng from the use of ths document.
3 Contents Introducton 5 2 Plant/solatmoshere nteractons 3 2. Prectaton Evaotransraton Multle cro coeffcent method Actual transraton Actual ondng and sol evaoraton Actual nterceton evaoraton from canoy Data summary and coulng to WOFOST 2 3 Sol water Introducton Pondng water and nfltraton Theory Imlementaton Unsaturated flow Theory Imlementaton Coulng to regonal groundwater flow Theory Imlementaton Data summary 5 4 Dranage Introducton Theory Model mlementaton Data summary 62 5 Surface water Introducton Theory Model mlementaton Schematzaton and hydraulc metafunctons for channel flow Dynamcs of channel flow Water on the sol surface Data summary 74
4 6 Water management Introducton Land use Urban areas Srnkled cros Surface water Wers Surface water suly lnks Dscharge ums 80 References 8 Aendx A Steadystate unsaturated flow smulatons 85 Aendx B Cro coeffcents as a functon of LAI 87
5 Suly caacty Aqutard surface water Surface water system Subsurface rrgaton or dranage Root zone Callary rse or ercolaton Phreat c level Introducton st Aquf er Aqutard nd 2 Aqufer Node ont fnte element grd Hydrologcal base All regonal hydrologc models suffer the defcency that the hydrologc cycle s not fully smulated n all ts rocesses and dmensons. In the frst lace ths concerns the satal lmtaton of a model, whch necesstates the makng of assumtons wth resect to external boundary condtons. Furthermore, most models cover only art of the rocesses wthn a regon. The remedy s to then ntroduce nternal boundary condtons. But these condtons are more often than not a gross schematzaton of realty, lackng essental feedback mechansms. To some degree, that even ales to the commonly used assumton that the rectaton and evaoratve demand of the atmoshere can be seen as external boundary condtons: the larger the basn, the less ths assumton s vald. For comng to grs wth many ssues of ntegrated water management, t s necessary to h a model code that covers the whole (regonal) system, ncludng lantatmoshere nteractons, sol water, groundwater and surface water. SIMGRO (a dated acronym of SIMulaton of GROundwater, Querner and Van Bakel, 989, Veldhuzen et al., 998) was develoed for that urose. The name SIMGRO was formerly used for referrng to an ntegrated model code, ncludng submodels for the comartments and rocesses as shown n Fgure. Now t s used n the meanng of a modellng framework. Ths framework has been connected to a number of nhouse comonents, but also has ossbltes for coulng to other codes. The nhouse comonents are ndcated n the form of chater numbers n the overall scheme of regonal water flows gven n Fgure 2. They can be groued as follows:  an SVATmodel that s commonly referred to as MetaSWAP, coverng the lantatmoshere nteractons and sol water;  a smlfed surface water metamodel;  a dranage ackage, for smulatng groundwater dranage wth fast feedback from surface water. Fgure Schematzaton n SIMGRO of the hydrologc system by ntegraton of saturated zone, unsaturated zone and surface water (Querner and van Bakel, 989). Alterrareort 93._V
6 6 srnklng 2 rectaton Interceton storage nterceto n 6 srnklng net rectaton/srnklng 3 Deresson storage 3 surface runoff evaotransraton nfltraton Sol mosture storage 3 ercolaton callary rse Groundwater Groundwater storage (mult layer) storage (mult layer) lateral flow subsurface rrgaton dranage 4 Surface water storage 5 surface water flow 5 Groundwater Groundwater storage (mult layer) storage Surface water storage Fgure 2 Schematzaton of regonal water flows by means of transmsson lnks and storage elements. The flows and lnks that can be modelled wth nhouse comonents are ndcated wth  references. For groundwater (MODFLOW) and surface water (SOBEKCF) there are connectors for external models. The current ossbltes for coulng to other codes are:  MODFLOW for groundwater;  SOBEKCF for surface water. SOBEKCF can be used n combnaton wth the smlfed surface water metamodel, wth the latter mlemented for the ustream waterways and SOBEKCF for the larger downstream ones. The current software mlementaton stll nvolves coulngs to secfc external codes. But the way n whch ths has been done makes t a relatvely small ste to arrve at a comonentbased mlementaton that allows connectons to any code that s OenMIcomlant (.e. comlant wth Oen Modellng Interface, HarmonIT, 2005). Ths software mgraton to an OenMIcomlant verson of SIMGRO s lanned for the near future. Alterrareort 93._V
7 Fgure 3 Examle of how the satal schematsatons of an ntegrated model can be constructed. The bottom layer nvolves the unts obtaned from an overlay of the land use and sol mas. The next layer reresents the cells of the groundwater model, followed by the subcatchments of the surface water model n the next layer. The to layer shows how the schematsatons h been combned. If there s just one colour n a groundwater cell, then there s just a sngle SVATunt lnked to t. If there are more colours, then two or more SVAT unts are connected. Connectons between models are n SIMGRO defned n terms of mang tables, as s done when usng OenMI technology. Each record of such a table contans a ar of dentfers that defne an dbased lnk between two models. In Fgure 3 an examle s gven of how a model schematsaton can be bult and then coded n the form of these dbased lnks. In the case that a grd cell n the to layer of the fgure contans more than one colour, then two or more SVAT unts are connected to a sngle groundwater cell. Ths s the socalled N: coulng. In the shown schematsaton there s also an N: coulng between the SVAT s and the subcatchments of the surface water model. SIMGRO has two otons for connectons between models:  a fluxbased lnk, qlnk;  a headbased lnk, hlnk. Fluxbased lnks are the tye generally used n combnaton wth OenMI technology. In such a lnk between Model A and Model B, the frst comutes an exchange flux. If ths flux s based on a sngle cycle of model nteracton, then the scheme s called exlct. Wth ths method two tyes of related roblems can occur:  the flux causes Model B to run dry, causng a water balance error f Model A stll assumes that the water s avalable;  the flux causes a large dsturbance of the head smulaton n Model B, whch n the next cycle leads to destablzaton of the model combnaton, nvolvng oscllatory behavour of the flux. Alterrareort 93._V
8 In order to avod water balance errors, a fluxbased lnk can be made to nclude a ut and a get oeraton. In the utste the Model A sends a demand to Model B n the form of a flux. That model then checks whether the water s avalable, and determnes the demand realsaton, whch s then returned n a ut by B. The realsaton s then cked u va a get by Model A. The destablzaton roblem can be solved by ncludng an teraton cycle. Such a cycle wll also remedy any water balance errors, f allowed to fully converge. In that case the coulng method s called mlct. In SIMGRO, the ut/get cycle s used for the fluxbased coulng between MetaSWAP and a surface water model. In the case of a N: coulng, the demands are frst totalzed. On return of the realsaton, ths s then dstrbuted by SIMGRO over the unts that laced the demand (Fgure 4). The teratve coulng s only used for a fluxbased lnk between MetaSWAP and MODFLOW. Sol water MetaSWAP column Tye hlnk: Tye qlnk: er teraton er teraton Recharge Bottom flux Srnklng Srnklng demand ut demand ut Storage coeffcent Per t sw Dranage Infltraton demand ut Per t sw Head Demand realzatons Per t sw Runoff/runon Srnklng demand ut Per teraton cycle Head SIMGRO Demand totalzer Realzaton dstrbutor Groundwater MODFLOW MODFLOW: Dranage Infltraton Surface water otons: SurfW SOBEKCF Fgure 4 Modules wth relatonshs and otons. MetaSWAP s the nhouse SVAT (SolVegetatonAtmoshere Transfer) model of SIMGRO. SurfW s the nhouse surface water metamodel. It can be used n combnaton wth the hydraulc model SOBEKCF. The lnks nvolve the uttng of demands and the rely n the form of a demand realzaton. The left half of the scheme has a tme ste of the groundwater model, t gw, the rght half of the fast rocesses, t sw A socalled hlnk s a secal feature of the SIMGRO framework, nvolvng the head as a shared state varable of two connected models. The shared varable s alternately udated by Model A and Model B. Ths s done n the followng two substes:  an exlct subste by Model A;  an mlct subste by Model B, usng nformaton obtaned n the frst subste. Alterrareort 93._V
9 In the second subste the assumton s that the fluxes of the frst subste reman unchanged. Under ths assumton, the second subste for Model B only affects Model A n terms of ts water storage. In order to take ths nto account n the second subste, the frst subste also rovdes the relatonsh between the shared state varable and the storage n Model A. The storage relatonsh s ncluded n the set of equatons that s to be solved n the mlct scheme of Model B n the second subste. In an exlct scheme the fluxes between the models are urely determned by the udate of Model A. But n the hlnk scheme the total fluxes between the connectng models are determned from budget calculatons after Model B has been udated. Ths s what dstngushes the method from a urely exlct one, and justfes callng t semmlct. For the MetaSWAPMODFLOW coulng the hlnk s the most used method, because t makes the hreatc level avalable to MODFLOW: the hlnk can be seen as a qlnk between the hreatc level and the MODFLOW head, wth a flow resstance that aroaches zero. The MODFLOW head can then be used as the hreatc level n the MODFLOW dranage ackages (RIV, DRN, etc.). For the MetaSWAPSurfW modellng the hlnk s used for the runoff/runon smulaton, because the low resstance of ths flow rocess would otherwse requre very small tme stes or a comutatonally demandng teratve mlct scheme to avod destablzaton of the model combnaton. As can be seen from the overvew n Fgure 4, SIMGRO has searate tme stes for the fast rocesses (Δt s ) and for the slow ones (Δt g ). The fast rocesses nclude lant/atmoshere nteractons, flow over the sol surface, dranage wth surface water feedback, and channel flow. Sol water and groundwater flow are modelled as slow rocesses. Tycal tme stes used n the current modellng ractce are Δt s = hour and Δt g = day. In the flow chart wth an overvew of the SIMGRO modellng cycles gven n Fgure 5, the fast tme ste s resent n the to half, and the slow ste n the bottom half. The teraton cycle around the groundwater model s needed because most groundwater models can not handle a nonlnear storage relatonsh. (By contrast, all surface water models h such a faclty.). In the software mlementaton of the coulng scheme, the recharge s assed to the groundwater model along wth the storage coeffcent, for each teraton cycle. But the recharge s n fact only udated once er groundwater tme ste by the MetaSWAP model tself. In the case of a qlnk, however, the bottom flux s udated for every teraton cycle. Alterrareort 93._V
10 Start Intalsaton of t s = t g t s = t s + t s Demand realsaton To system rocesses er SVATunt:  atmosherc nteractons  nterceton, throughfall  cro water utake from root zone  ondng, nfltraton, surface runoff  dranage/nfltraton wth sw feedback Surface water rocesses er trajectory:  levels and flows  sules to to system No Totalzatons over gw tme ste  recharge of sol/groundwater dranage/nfltraton t s = t g + t g? t g = t g + t g Unsaturated rocesses er SVATunt:  ressure heads  ercolaton/callary rse No Formulate groundwater rocesses er cell:  resdual sol/groundwater recharge  tmevarant storage coeffcent  dranage wth gw feedback No Groundwater rocesses er cell:  heads and flows  sules to to system Convergence? Fnalze to system:  water content of sol  storage on sol surface Sto Yes t g = t end? Fgure 5 Overvew of modellng cycles. The tme varable of the fast tosystem rocesses t s s synchronzed wth t g of the groundwater rocesses at the begnnng of each groundwater tme ste. The resdual sol/groundwater recharge s the remanng column recharge after dscountng the water needed for udatng the ressure heads. Water management s not shown. Alterrareort 93._V
11 The smulaton of (ossble) changes n water management s the goal of most studes. Gven ts crucal mortance for the ractcal relevance of the model, the management otons are descrbed n a searate secton ( 6). But of course the dstncton between what s consdered water management and what natural functonng deends on the ersectve that one has. Here any tye of landuse nvolvng vegetated sol s consdered to be natural, even though t s some form of agrculture. In the case of landuse, the term water management s reserved for d areas, nvolvng some form of nterference n the cycle, e.g. urban areas. The layout of manmade dtches s descrbed as f t were art of the natural system, whereas t s of course not. In the case of surface water the term management s reserved for the manulaton of the water flows and levels through the use of structures. In the followng sectons a descrton s gven of the used concetualzatons, followed by more detaled descrtons about the way they h been mlemented. To ad the reader n quckly locatng the descrtons, the overvew n Fgure 2 has been furnshed wth hyerlnks to the subsectons. Alterrareort 93._V7.2.25
12 Alterrareort 93._V
13 2 Plant/solatmoshere nteractons 2. Prectaton It s well known that the measurement of rectaton usually contans systematc errors due to wndeffects. The sze of the error s usually n the order of a few % (<5%), but should be accounted for. The natural rectaton can be augmented by srnklng. The total (gross) rectaton s then gven by: P g ( s, n t) P( t) P ( t) () where: P g (t) = gross rectaton rate at tme t (m 3 m 2 d  ) P(t) = natural rectaton rate at tme t (m 3 m 2 d  ) P s,n (t) = net srnklng rate at tme t (Eq. 06) (m 3 m 2 d  ) In the above relatonsh (and n others for the rocesses at or around the sol surface) we use the dmenson [L 3 L 2 ] when the value eruntofarea s meant. After multlcaton by the relevant area fracton the satal mean s obtaned n [L]. Not n all cases ths dstncton s relevant; we then use the latter notaton. The tme deendency of the rectaton (and other varables reresentng atmosherc boundary condtons) s raged over the used smulaton nterval. For ths the nterval t s of the 'fast' rocesses s used. The use of tmeraged values s ndcated va the suerscrt ''. So Eq. then becomes: P g P P (2) g s,n where: P g = gross rectaton rate, tme raged for nterval (m 3 m 2 d  ) P = natural rectaton rate, tme raged for the nterval (m 3 m 2 d  ) P = net srnklng rate, tme raged for the nterval (m 3 m 2 d  ) s,n Incomng rectaton (and/or srnklng water) ether falls drectly on the ground surface as free through fall or s nterceted by the vegetaton canoy. Part of that nterceton drectly slashes off at the mact. The art that does not, s subsequently stored on the canoy and evaorate, or else reach the ground surface as stem flow or dros from the les. The nterceton of ranfall s modelled as a dffuse vertcal rocess, wth no clearcut dvson between vegetaton cover and bare sol. Prectaton that comes nto contact wth the canoy  and that does not drectly slash off  s denoted as the gross nterceted rectaton. It s modelled as a smle fracton of the gross rectaton: P, g ( t) c Pg ( t) (3) Alterrareort 93._V
14 where P,g (t) s the gross nterceted rectaton as a functon of tme (m d  ), P g (t) s the gross rectaton (ranfall lus srnklng rrgaton) (m d  ), and c s the nterceton fracton. Currently t s set equal to the lght fracton that s nterceted. The smulaton of the nterceton evaoraton from the canoy s descrbed below. 2.2 Evaotransraton 2.4. Multle cro coeffcent method An evaotransraton smulaton method should be accomaned by a method that rovdes accetable arameters for the ntended user communty. In our case we mostly use the cro coeffcents gven by Feddes (987) for agrcultural forms of land use. Those coeffcents, however, are ntended for an all n one evaotransraton smulaton method that does not account for varable feedbacks from cro growth. Those feedbacks affect each of the evaotransraton comonents n a secfc manner. We therefore model evaotransraton as four searate and artly nterdeendent terms: cro transraton, canoy nterceton evaoraton, sol evaoraton and ondng evaoraton. The arameters are calbrated usng Feddes (987). As startng ont, we used an adated form of the dual cro coeffcent aroach gven by Wrght (982) and Allen et al. (2005): T E ( K K ) ET (4) s, cb ew 0 where T s the otental cro transraton (m d  ), E s, s the otental sol evaoraton (m d  ), K cb s the basal cro coeffcent (), K ew s the evaoraton coeffcent of a wet bare sol that s artly shelded by vegetaton (), and ET 0 s the reference cro evaotransraton (m d  ). The reference cro evaotransraton can be calculated wth the PenmanMonteth method (Wrght, 982; Allen et al., 998) or wth the smlfed Makknk equaton, descrbed by De Brun (98, 987): s K ET s L v c where s s the sloe of the vaour ressure curve, γ s the sychrometrc constant (kpa K  ), K s the ncomng global radaton (W m 2 ), L v s the latent heat of vaorzaton of water (J kg  ), and c s a converson factor (m d  / kg m 2 s  ). De Brun (987) gves the followng reasons for choosng the Makknk equaton: a. Its behavour s very smlar to that of the Penman formula; b. It s remarkably smle: t requres only ar temerature and global radaton as nut, c. Under dry condtons Makknk s formula erforms better. (5) Alterrareort 93._V
15 Sol evaoraton also ncludes the evaoraton from sol that s artly shelded by a canoy. The bare sol s assumed to be dffuse, meanng that we do not dstngush between sol that s covered by vegetaton and sol that s not, n vew of the varable angle of ncomng radaton. We assume that the net radaton nsde the canoy decreases accordng to an exonental extncton functon of the leaf area ndex (LAI) and that the sol heat flux can be neglected (Goudraan, 977; Belmans, 983): K ew LAI gr e K (6) ew00 where gr s the extncton coeffcent for solar radaton (), LAI the leaf area ndex, and K ew00 the evaoraton coeffcent of a wet sol recevng full radaton. Rtche (972) and Feddes (978) used κ gr = 0.39 for common cros. More recent aroaches estmate κ gr as the roduct of the dmensonless extncton coeffcent for dffuse vsble lght, κ df, whch vares wth cro tye from 0.4 to., and the extncton coeffcent for drect vsble lght, κ dr : κ κ gr df κ dr (7) If the sol s nundated due to ondng, then the ondng evaoraton s of course the domnant rocess and the bare sol evaoraton s suressed. The otental evaoraton rate of ondng water s calculated wth a cro coeffcent n a fashon that s analogous to that for bare sol. The calculaton of the actual ondng and sol evaoraton s gven n A wet canoy leads to an ncreased evaotransraton due to nterceton evaoraton, whch lowers resstance for the mosture flow from the canoy to the atmoshere. Wrght (982) accounts for ths lower resstance by adjustng the cro and bare sol coeffcents. We model these effects searately, va the mentoned sol evaoraton method and by exlctly modellng canoy nterceton evaoraton. Interceton evaoraton from a vegetaton canoy s not senstve to sol mosture condtons. Ths makes t relevant to model t searately from the transraton. The smulaton of the actual nterceton evaoraton s descrbed n Evaoraton from a wet canoy s assumed to be a domnant rocess: as long as t s actve, the transraton s assumed to be zero. Ths assumton s based on the notons that evaoraton has frst choce for usng the energy flux and that due to a hgher ar mosture content the transraton s suressed. To model canoy nterceton evaoraton and ondng evaoraton as domnant rocesses, Eq. 4 s exanded to: T E E E [ K (W ) K K (W ) K ] ET (8), s, ond, cb frac, w ew frac,ond ond 0 where E, s the otental canoy evaoraton rate (m d  ), and K w s the evaoraton coeffcent of a wet canoy (), wth K w > K cb, E ond, s the otental ondng evaoraton rate (m d  ), K ond s the cro coeffcent of ondng evaoraton (), W frac, s the tme fracton that nterceton evaoraton s actve () and W frac,ond s the tme fracton that ondng evaoraton s actve (). The value of K ond s reduced for the fracton of lght that reaches the ondng water, n a smlar fashon as s done for a wet sol (converson of K ew00 to K ew ). Alterrareort 93._V
16 The used tme fracton that nterceton evaoraton s actve s comuted as E Wfrac, (9) E,a, where E,a s the actual canoy nterceton evaoraton rate (m d  ), raged over the smulaton tme nterval (m d  ), and E, s the tmeraged otental value (m d  ). The tme fracton that ondng evaoraton s actve s comuted as: W frac, ond E E ond, a ond, (0) where E s the actual ondng evaoraton rate (m d  ), tme raged over the ond, a smulaton tme nterval (m d  ) and E s the tmeraged otental value (m d  ). ond, Actual transraton In the current model mlementaton the otental canoy evaotransraton s unformly dvded over the root zone deth, as s done n Feddes et al. (978). Prasad (988) would be an alternatve: nstead of a unform dstrbuton the rootng ntensty decreases lnearly wth the deth. For the effect of lmtng sol mosture condtons on the evaotransraton the functon defned by Feddes et al. (978) s used: T α T () a E where: T a = actual evaotransraton rate (m d  ) T = otental canoy transraton rate (m d  ) α E = sol mosture reducton factor () T low.0 E low T low α E() TE hgh 4 3l 3h 2 (m) Fgure 6 Reducton coeffcent for root water utake, α E, as a functon of sol water ressure head and otental transraton rate T (see T hgh and T low ) (after Feddes et al., 978) 0.0 Alterrareort 93._V
17 Water utake by roots s zero when the sol water ressure head s below 4, whch s assumed to be the wltng ont. Sol water ressure head 3 s called the reducton ont. In between the wltng ont and the reducton ont the evaotransraton rate s lnearly reduced. The reducton ont deends on the otental evaotransraton (see T low and T hgh ). Between ressure heads 2 and 3 the evaotransraton s at ts maxmum ('otental'). Alcaton of the reducton functon for < 2 to the mean root zone ressure head (see 3) would yeld erroneous results due to the nonlnear form of the functon. For ths reason the root zone ressure head s frst downscaled to 0 equalszed sublayers of the root zone. The functon s then aled to the searate sublayers. As a result of oxygen defcency n the root zone, water utake s hamered for many cros between 2 and. The reducton coeffcent s zero for >. But n we do not aly the reducton for > 2 to the searate sublayers of the root zone lke was roosed by Feddes et al. (978). Alcaton to the searate sublayers s consdered to be nonrealstc because t assumes that the vegetaton has no means to comensate for oxygen defcency n art of the root zone. Esecally under wet condtons wth thck root zones (of trees) the extracton attern wll vary accordng to the season: unform over the whole root zone n stuatons wth a low groundwater level, and only n the uer arts of the root zone n stuatons wth a shallow groundwater level. For we use the ressure head n the to art of the root zone (to 0%) for comutng the reducton coeffcent that s then aled to all of the layers of the root zone n a unform manner Actual ondng and sol evaoraton Actual ondng evaoraton s modelled wth a smle bucket model that evaorates at the otental rate (K ond ET 0 ) as long as there s water on the sol surface. The dynamcs of ondng water are descrbed n 3.2. The bare sol evaoraton s a rocess nvolvng the very thn uer sol crust layer. When the layer dres out, a selfmulchng effect occurs, lmtng further loss of mosture. Modellng of the sol water transort at such a detaled scale s beyond the scoe of a metamodel. Instead, we use the method of Boesten and Stroosnjder (986). The bare sol evaoraton for a tme ste s derved from a smle model for the cumulatve values of actual and otental evaoraton snce the start of the dryng erod. The model s centred around the followng assumed relatonsh: ΣE a ΣE a ΣE 2 ΣE for for ΣE ΣE (2) where: β 2 emrcal sol arameter, wth a default value of (m ½ ) ΣE a sum of actual evaoraton snce the start of the dryng erod (m) ΣE sum of otental evaoraton snce the start of the dryng erod (m) Alterrareort 93._V
18 The arameter β 2 determnes the length of the otental evaoraton erod, as well as the sloe of the ΣE a versus (ΣE relatonsh n the sol lmtng stage. If there s not an excess n ranfall (P n < E ), the udate of ΣE s done wth: j j ( ΣE ) (ΣE ) ( E Pn ) (3) where P n s the net rectaton that s equal to the drng rate D of the nterceton model, where suerscrt j s for the tme level. Subsequently (ΣE a ) j+ s calculated from (ΣE ) j+ wth Eq. 2 and E a for the tme ste s calculated wth: j j E a Pn ( ΣEa ) (ΣEa ) (4) If there s an excess n ranfall (P n > E ) t can be assumed that the actual evaoraton wll equal the otental value: Ea E (5) The excess ranfall artly (or fully) reverses the dryng out of the sol. To take ths nto account the excess ranfall s subtracted from the cumulatve ΣE a : j j ( ΣEa ) (ΣEa ) ( Pn E ) (6) If ths udate yelds a negatve value for ΣE a t s set to zero, sgnallng a new start for the dryng rocess. In order to mantan consstency n the model the new ΣE s obtaned from the nverse form of Eq. 2, wth theσe a term n the rghthand sde of the rearranged equaton Actual nterceton evaoraton from canoy The nterceton water balance s smulated wth: ds ( t) dt P ( t) E ( t) D ( t) (7),g where S (t) s the water stored n the nterceton reservor (m), E s the canoy evaoraton rate (m d  ), and D (t) s the canoy drng rate (m d  ). It s essental to use an analytcal soluton to the above equaton, to avod numercal errors caused by tme dscretzaton. That s then avoded f the tme ste s not larger than the gaugng nterval of the rectaton and the evaoratve demand. Drng from the canoy starts when the canoy becomes saturated, any excess water drectly drs through. The storage of water n the canoy s assumed to be bounded: S t) S ( t) (8) (, ca Alterrareort 93._V
19 where S (t) s the water stored n the nterceton reservor (m), and S,ca (t) s storage caacty of the nterceton reservor (m). The canoy evaoraton rate s assumed to deend on the canoy saturaton and the otental evaoraton rate as: S ( t) E ( t) [ f,mn ( f,mn ) ] E, ( t), for S ( t) 0 S (9) E ( t) 0, for S ( t) 0,ca where E, (t) s the otental evaoraton rate of a wet canoy (m d  ), and f,mn s the startu fracton of the canoy evaoraton reducton factor (). If f,mn = 0 the relatonsh s lnear lke n Rutter (97). For a value of.0 the evaoraton rate equals the otental rate as long as there s water n the reservor. The method to obtan the otental rate E, (t) s gven n Eq. (8). In the soluton scheme, the tme varaton of the rectaton rate s handled as a stewse functon. Ths s denoted by the suerscrt. The nonlnear canoy evaoraton rate relatonsh (Eq. 9) requres searate treatment for f,mn equal to unty. Case : Dscrete deendence of evaoraton on canoy saturaton For f,mn equal unty (Eq. 9a), the followng water balance can be made: S S j j S j ( P mn( S,g j, S E j,ca, ) ; ) t S j max( S j,0) (20) j where S (m) s the water stored n the nterceton reservor at tme level j, P,g, (m d  ) s the gross nterceted rectaton rate, tmeraged rate over nterval t, and t (d) s the length of the smulaton tme nterval. In case that the reservor does not become full, the evaoraton follows from the balance: E j j S S )/ ( t P,g (2) where E (m d  ) s the tme raged canoy evaoraton rate. In ths case the drng rate s zero. In case the reservor becomes full, the smulated evaoraton rate equals the otental value durng the whole nterval owng to the dscrete relatonsh of the canoy evaoraton (otental value for a nonzero storage) and the use of tmeraged values: E E (22), The net nterceted rectaton and the dr to the sol surface can now be found from: P,n D ( S P j g S  P j,n )/ t E where P,n (m d  ) s the tme raged net nterceted rectaton and D s tme raged the drng rate. The equatons above can also be used n case the otental canoy evaoraton rate equals zero. (23) Alterrareort 93._V
20 Case 2: Semcontnuous deendence of evaoraton rate on canoy saturaton If f,mn n Eq. 9a s less than unty, there s a semcontnuous deendence of the evaoraton rate on the canoy saturaton. After nsertng the exresson for the evaoraton rate as gven n Eq. 9a nto Eq. 7 and rearrangng, we get for an unsaturated canoy (S <S,ca, D =0): ds ( t) E, βs ( t), wth ( f,mn ) and P,g f,mn E, (24) dt S Assumng that β s nonzero (see the above Case 2 for β = 0) and that the nterceton reservor does not become full (S < S,ca, D c =0), the soluton to the dfferental equaton for S (t) s gven by: j j t S ( S )e (25) S j max( S j,0) If the reservor does not become full (S <S,ca ) the evaoraton s now gven by Eq. 2. If S (Eq. 25) yelds a value >S,ca then we frst determne the tme at whch ths haens by solvng for t ca ( t ca t): j t ca S,ca ( S )e (26) j tca ln[( S ) /( S,ca )] where t ca s the tme to fll the reservor durng whch there s no drng from the canoy. The canoy evaoraton can thus be found from a balance lke the one gven n Eq. 2; for the remanng art of the nterval ( t  t ca ) the evaoraton rate s equal to the otental value. So the total canoy evaoraton can be found from:,ca E j j ( ) S tca P,g t tca E, [ S t ] (27) The drng rate then follows from Eq. 23. Alterrareort 93._V
21 2.5 Data summary and coulng to WOFOST For the smulaton erod, the tme seres nformaton of the meteorologcal condtons should be avalable n the form of a stefuncton wth a tme nterval of t s. That concerns both the rectaton and the data needed for smulatng the evaotransraton. For the Makknk method only the values of the reference evaotransraton are needed. If the PenmanMonteth method s used the followng data are needed:  radaton;  temerature;  humdty;  wnd seed. These data can be suled for more than one gaugng staton n the regon. In that case the relevant staton should be secfed for each of the SVATunts of the sol water /groundwater model. A method for ntroducng extra satal varaton s to secfy calbraton factors for each of the SVAT s. It s also ossble to secfy all of the tmedeendent data va grds coverng the satal extent of the model area. If the data are only avalable on a daly bass, then some form of downscalng should frst be aled for smulatng the nterceton evaoraton wth some degree of accuracy. Per SVATunt the domnant landuse should be secfed. The arameters of the transraton reducton functon are requred for all methods, as are the lghtextncton coeffcents. For modellng the nfluence of the vegetaton on the evaotransraton arameters two methods are avalable:  smle vegetaton model, wth the cro develoment (and deendent arameters) gven er day of a calendar year n a fle;  advanced vegetaton model, wth the dynamc smulaton of the cro develoment wth WOFOST (Van Keulen and Wolf, 986; Van Deen et al.,989; Sut et al. 994). WOFOST and MetaSWAP (Fgure 7) communcate on a daly bass. Cro water stress causes the leaf ores to close artly or comletely, thus mnmzng mosture loss. Ths also reduces the CO 2 assmlaton and cro growth. To model ths reducton, WOFOST requres the relatve transraton as nut,.e. T a /T. Durng erods of nterceton evaoraton the transraton s assumed to be zero; then the relatve transraton s set equal to the α rw coeffcent for a low value of T. WOFOST returns the root zone deth and LAI; the latter s subsequently used to calculate:  the fracton of radaton reachng the sol surface;  the transraton cro coeffcent;  the ranfall nterceton fracton, the canoy nterceton storage caacty, and the nterceton evaoraton cro coeffcent. The LAI affects the nterceton n varous ways: Alterrareort 93._V
22  the fracton of ranfall that drectly reaches the sol surface wthout assng through the nterceton reservor s set equal to the fracton of radaton reachng the sol surface;  the nterceton storage caacty of a canoy s assumed to be lnearly deendent on LAI accordng to S,ca = s,ca LAI, where s,ca s the caacty er unt of LAI (mm LAI  );  the nterceton evaoraton cro coeffcent K w s set to.2 tmes K cb ; ths factor has been set slghtly lower than the rato between the coeffcents for oen water and short grass (.25). The manner n whch WOFOST s used for dervng relatonshs based on LAI s descrbed n Aendx B. Also descrbed s the way n whch the tables for the smle vegetaton model are derved from a 30year run wth the couled MetaSWAP WOFOST models. Fgure 7 Coulng MetaSWAPWOFOST as mlemented n the SIMGRO modellng framework Alterrareort 93._V
23 3 Sol water 3. Introducton As ndcated n the ntroductory chater, water on the sol surface has a multle dentty. Here we descrbe the manner n whch t takes art n the sol column rocesses, ncludng the nfluence of atmosherc nteractons descrbed n the revous secton. For the model formulaton we use a control box 0 that extends uwards from the sol surface. The subsurface sol water dynamcs are descrbed usng two control boxes: for the root zone, the shallow subsol and the dee subsol (Fgure 8). We descrbe the unsaturated zone model n detal n 3.3. The water balances and smulatons are made at the aggregate scale of the control boxes, but the model n fact has contnuous mosture rofles on the background, meanng that at any desred moment the detaled head and mosture content rofles can be made avalable f needs be. Ths dstngushes the aroach from models based on lumng of the vertcal doman. q run P n E T q nf Root zone: control box q,r q c,r Subsol: control box 2 q,2 q c,2 Phreatc level Fgure 8 Flows n the sol water model. Exlanaton of symbols: P n net rectaton; q nf nfltraton; q run  surface runoff; E evaoraton of bare sol; T transraton; q ercolaton; q c callary rse. Alterrareort 93._V
24 3.2 Pondng water and nfltraton 3.2. Theory Water stored on the sol surface has dverse functons and nteractons. Here we confne ourselves to ts role as a reservor and as an ntermedate source for sulyng nfltraton water. The storage on the sol surface s schematzed nto two fractons:  mcro storage;  macro storage. The mcrostorage s formed by small deressons n the sol surface at subgrd scale. When after an nundaton the land falls dry the water n the mcrostorage s retaned. The water n macrostorage can freely move over the sol surface, restrcted only by natural barrers and constructons n the surface water channel network. The schematzaton of the surface water system can be extended to the sol surface, obtaned from an GISanalyss of the sol surface toograhy. In that case the barrers consst of the saddle onts n the sol surface landscae. Pondng water that s macro storage on the sol surface has a multle dentty. The stes nvolved n the multle dentty concet are descrbed n and llustrated n Fgure 29 through Fgure 33. The modellng of nfltraton s notorously dffcult for varous reasons, of whch we menton:  the role of crustformng at the sol surface;  the comlexty of the rewettng rocess of a dred out sol;  the role of subgrd scale flow rocesses over the sol surface; water that becomes runoff at one sot can after all nfltrate n a small deresson just a few metres away. We do not attemt to model these asects va a formulaton n terms of a dfferental equaton. Instead we follow a ragmatc aroach nvolvng a mnmum of arameters that stll allow enough freedom for the modeller to control the smulaton along broad lnes. A recent addton (204) s the oton of usng the Curve Number method (USDA SCS, 972). The amount of water n the ondng reservor ( ondng storage ) reacts drectly to ncomng rectaton and exchange wth the surface water model; thus we model the dynamcs of the ondng reservor wth the tme ste of the fast rocesses. The use of a dfferent tme stes for the fast and slow rocesses can lead to numercal artefacts n the model f no recautons are taken. These recautons can take on two dfferent forms:  sreadng of the effect of the revous udate of the groundwater model on the new seres of fast tme stes;  antcaton on the behavour of the model durng the next udate of the groundwater model n reacton to fluxes and state varables of the fast rocesses. Alterrareort 93._V
25 3.2.2 Imlementaton At the begnnng of the frst tme cycle for the fast rocesses wthn a new groundwater tme ste (see Fgure 5) the groundwater above sol surface (f resent) s converted nto ondng water: S j0 ond max{ 0; h j0 h ss } (28) where: j 0 S = ondng storage at the begnnng of the frst fast tme cycle wthn the ond groundwater tme ste (m) h j=0 = groundwater level at tme level j =0 of the fast rocesses,.e. at the end of the revous groundwater tme ste (m) h ss = elevaton of the sol surface (m) The stuaton wth uward seeage to the sol surface s handled n a secal manner. The above equaton s then mlemented n a modfed form, n whch the nfluence of the uward seeage s sread out over all of the fast tme stes wthn the consdered groundwater tme ste. Wthout ths modfcaton the model would roduce artfcal eaks n the surface runoff durng the frst fast tme ste. Infltraton can of course only occur f there s water avalable n the form of net rectaton and/or water stored from the recedng erod. We gve nfltraton of ondng water rorty over evaoraton. So n order to determne the water avalablty for nfltraton the amount of water n the ondng reservor s frst udated wth: S' j+ ond = S j ond + P n Δt s (29) where: j S ond j ond = ondng storage at tme level j of fast rocesses (m) S' = ondng storage after ntermedate udate for net rectaton (m) P n = net rectaton rate, tme raged (m d  ) Δt s = tme ste of fast tme cycle (d) The uer bound for the avalablty of nfltraton water s then gven by: q nf, max,ond S / Δt (30) j ' ond s where q nf,max,ond s maxmum avalable nfltraton rate from ondng water (m d  ) For stuatons wth free ercolaton to the groundwater we use a smle relatonsh for determnng the maxmum nfltraton rate at the sol surface q nf,max,s : q j + nf, max,s qnf,bas + S' ond/ = c to,down (3) where: q = maxmum nfltraton rate at sol surface (m d  ) nf,max,s Alterrareort 93._V
26 q nf,bas = basc nfltraton rate for stuatons wthout ondng (m d  ) c to,down = vertcal flow resstance n to of sol (d) The nfluence of the frst term can be cancelled by secfyng hgh values of q nf,bas n the nut fle. The use of the second term nvolvng c to,down s otonal. In fully saturated condtons and wth a secfed c to,down the nfltraton rate s found from the head dfference between ondng water and groundwater dvded by c to,down. In stuatons wth the head of the groundwater hgher than that of the ondng water the resstance for uward flow s used c to,u. If the resstance s not secfed, then we assume freedom of movement of water through the sol surface n stuatons that are fully saturated. Infltraton can also be hamered by a bottleneck further down n the system,.e. n the sol water and/or groundwater system. Let us frst consder the case wth the groundwater level below the bottom of the root zone. The maxmum amount of storage sace avalable for nfltraton s then estmated wth: S q j =0 nf, max,r = Sr,sat  Sr +( Ea + Ta  nf,max,r = S nf,max,r /Δt s I + q,max )Δt g (32) where: S = avalable storage sace for nfltraton, based on root zone balance (m) nf,max,r S = saturated water content of root zone (m) r,sat j 0 S = water content of root zone at the begnnng of the frst fast tme cycle, r.e. the content at the end of the revous groundwater tme ste (m) E = sol evaoraton of revous fast tme stes,tmeraged over Δt g (m d  ) a T a = transraton of revous fast tme stes, tmeraged over Δt g (m d  ) I = nfltraton durng revous fast tme stes, tme raged (m d  ) q,max = maxmum ossble ercolaton rate (>0) to the groundwater (m d  ) q = maxmum nfltraton rate, based on root zone balance (m) nf,max, r Δt g = tme ste of sol water / groundwater submodels (d) The above exresson takes nto account that evaoraton and/or transraton from the root zone makes storage sace avalable for nfltraton water. The values gven n the above exresson are stll n the rocess of beng udated, durng each fast tme cycle. The tmeraged values are used because t s the volume that determnes the nfltraton sace. The maxmum ossble ercolaton rate to the groundwater s derved from the sol hyscal metafuncton q m gven n The ercolaton can be hamered by the stuaton n the subsol. For nvestgatng ths otental bottleneck a balance s made for the whole column: S q nf,max,c nf,max,c S ( E r,sat a S T max{ S j0 r a nf,max,c S I 2,sat / t s S Q,0} j0 2 sgw Q d G ) t g (33) Alterrareort 93._V
27 where: S = maxmum storage sace for nfltraton, based on column balance (m) nf,max,c S 2,sat = saturated water content of shallow subsol (box 2) (m) j 0 S = water content of root zone at the begnnng of frst fast tme cycle (m) 2 sgw Q = extracton for srnklng from groundwater (from frst layer) (m d  ) Q d = dranage / nfltraton (+ = to the column) (m d  ) G = regonal groundwater flow (+=to the column), last known value (m d  ) q = maxmum nfltraton rate, based on column balance (m) nf,max,c The nfltraton s then found from: q mn( q ; q ; q ; q ) (34) nf = nf,max,ond nf,max,s nf,max,r nf,max, c The next udate of the water stored n the ondng reservor s made wth: S" S' q t j ond j ond nf s (35) where: j S" = ondng storage water after udate for nfltraton (m) ond j The value of S" ond s used for determnng the maxmum avalablty of ondng evaoraton water. Subsequently the amount of water n storage s udated wth: S" E t j j ' ond S" ond ond s (36) where: S "' j = ondng storage after udate for evaoraton (m) ond E ond = ondng evaoraton rate (m d  ) The way the runoff s comuted deends on the used combnaton of modellng otons.  the choce whether or not the Curve Number method (USDA SCS, 972) s used;  the choce of method for flow over the sol surface: the resstance method or the ntegrated method nvolvng zero resstance. The Curve Number can only be aled n combnaton wth daly ranfall data. It has been mlemented followng Netsch et al. (20), ncludng the nfluence of the sloe wth the amendment of Wllams (995). In our techncal mlementaton, the runoff that s smulated wth the Curve Number method s converted to extra ondng by reducng the nfltraton. The ondng s then converted to runoff wth the algorthm gven further below. However, deendng on the method used for the flow over the sol surface, t s ossble that not all ondng water becomes runoff durng the consdered tme ste. Ths ondng water wll be a source for nfltraton durng the next tme ste, and thus cometes wth the rectaton for the avalable nfltraton sace. It (after converson to a rate by dvdng through the tme ste of fast rocesses) s therefore added to the gross ranfall for calculatng the Curve Number runoff of the next tme ste. Alterrareort 93._V
28 The Curve Number method s based on daly ranfall, but the runoff tself usually occurs durng a short tme san. So t s not realstc to smulate ondng evaoraton as f the runoff and ondng occur evenly sread over the whole day. Therefore the method s aled after the ondng evaoraton has been smulated. The converson of nfltraton to ondng water takes nto account that the nfltraton can already be lmted by condtons wth near saturaton: S j j "' + + ond= S"' ond +mn( qrun.cn, qnf )Δt s (37) where: q = runoff smulated wth the Curve Number Method (m d  ) run,cn For the flow over the sol surface, the resstance method has the advantage that the tmedelay due to the flow rocess s modelled to some extent. The dsadvantage s that f a low resstance has been secfed the smulaton can become unstable f used n combnaton wth a surface water model, and oscllatons can result. The ntegrated method s uncondtonally stable; the drawback s the mmedacy of transfer of water, whch can be unrealstc. In both methods, runoff can only take lace f the ondng level s above a threshold value. The storage caacty below ths threshold s termed the caacty of mcro ondng storage. If the resstance method s used, then for stuatons where there s no nfluence of the surface water level, the runoff s smly comuted wth: q run =[ S ond, mcro,ca S"' j+ ond ]/ c run (38) where: q run = runoff of the SVATunt (+ = to the column) (m d  ) S ond,mcro,ca = caacty of mcrostorage (m) = runoff resstance (d) c run For stuatons wth the surface water level above the runoff threshold, the threshold s relaced by the surface water level (relatve to the sol surface). For stuatons wth runon and the ondng level below the threshold, the ondng level s relaced by the elevaton of the threshold. If the ntegrated method used for the runoff smulaton (see and Fgure 29 through Fgure 33), the ondng water n the macrostorage s transferred to the surface water model. For ths the storage s frst decomosed nto the two tyes of storage wth: j j S" ' ond, mcro mn{ S"' ond; Sond, mcro, ca} (39) S "' j ond, macro S"' j ond S" ' j ond, mcro where: S " j = ondng mcrostorage (m) ' ond, mcro S = ondng macrostorage (m) "' j ond, macro Alterrareort 93._V
29 The surface water model then comutes hydraulc heads for the channel trajectores and the assocated subcatchments; each subcatchment conssts of one or more SVATunts of the to system model. After comletng the smulaton of surface water flow rocesses the newly determned levels are lnked to the resectve SVATunts wthn the subcatchments. The modfed amount of water n macrostorage s comuted by: j S" " ond, macro max{ hs ( hss Sond, mcro, ca ); 0} (40) where: j S = ondng water n macrostorage after surface water udate (m) h s h ss "" ond, macro = surface water level (m) = elevaton of the sol surface (m) and the total storage n the ondng reservor s comuted as S "" (4) j j j ond S"' ond, mcro S" " ond, macro where: j S = ondng water after udate for surface water flow (m) "" ond By comarng ths storage to the amount that was transferred to the surface water model, the net runon/runoff of the SVATunt can be calculated (>0 for runon): q j j run S"" onds"' ond] / [ t s (42) where: q run = runon/runoff of the SVATunt (m d  ) If the consdered fast tme ste nvolves an ntermedate udate wthn the (longer) j groundwater tme ste, the fnal ondng storage S ond s set equal to S "" j ond. The smulaton then contnues wth the next cycle startng from Eq. 29. If the smulaton has arrved at the end of the last fast tme cycle t s followed by a fnalzaton rocedure. In the case that there s hydraulc contact wth the groundwater the ondng storage s comared to that at the begnnng. The dfference s added as recharge to the groundwater model: q jn j0 recha, ond S"" ondsond) / q recha,ond j 0 ond ( t g where: = recharge rate of the vsble groundwater, tme raged over Δt g (m) S = ondng storage at begnnng of groundwater tme ste (m) S" " = ondng storage at end of last fast tme ste n, n = Δt g /Δt s (m) j n ond (43) j n After the groundwater model udate has been erformed, the ondng storage ond s fnalzed by alyng Eq. 28 for the new groundwater level. At ths ont the nfltraton rate can also be fnalzed on the bass of a water balance analyss nvolvng the fnal value of G, the regonal groundwater flow. S Alterrareort 93._V
30 Alterrareort 93._V
31 3.3 Unsaturated flow 3.3. Theory The model schematzaton assumes that unsaturated flow takes lace wthn arallel vertcal columns, wth each column connectng to a smulaton unt of a groundwater model. The hreatc surface acts as a movng boundary between the flow domans of the sol column models and the groundwater model. All lateral exchanges are assumed to take lace n the saturated zone. The man dea of the modelng method for the (unsteady) unsaturated flow s to use steadystate solutons to Rchards equaton as buldng blocks of a dynamc model, a socalled quas steadystate model. The arorate buldng blocks are for each tme level selected on the bass of water balances at the aggregate scale of control volumes for the root zone and the subsol. Put n mathematcal terms, the artal dfferental equaton for the unsteady flow (Rchards equaton) s relaced by two ordnary dfferental equatons: one for the varatons n the vertcal column (usng the steadystate form of the flow equaton) and the other for the varatons n tme (usng a water balance at aggregate scale). The descrbed method s a radcal redesgn of the one resented by de Laat (980) and Wesselng (957). A valdaton of the current method s gven n Van Walsum and Groenendjk (2008) Steady states For onedmensonal flow n an unsaturated sol wth root water extracton, the steadystate form of the flow equaton can be wrtten as: d dψ K( ψ) τ ψ z z h z z (, ) 0, 0, (44) d d subject to the boundary condtons ψ ( h) 0 (45) dψ K( ψ) dz z 0 q(0) where z = elevaton coordnate, taken ostvely uward (zero at the sol surface) (m) h = groundwater elevaton (m) ψ = ressure head (m) K(ψ) = hydraulc conductvty as a functon of ressure head (m d  ) q(0) = flux densty at the sol surface, taken ostvely uward (m d  ) τ(ψ,z) = deth and headdeendent extracton term for root water utake (m 3 m 3 d  ) A steadystate rofle s obtaned by secfyng the conductvty arameters of each dstngushed sol layer and by solvng Eq. 44 subject to mosed values for the groundwater elevaton h (Eq. 45), the otental flux densty at the sol surface q ot (0) (Eq. 46), and the otental total root water utake rate T ot of the root zone. A flexble root (46) Alterrareort 93._V
Confidence intervals for weighted polynomial calibrations
Confdence ntervals for weghted olynomal calbratons Sergey Maltsev, Amersand Ltd., Moscow, Russa; ur Kalambet, Amersand Internatonal, Inc., Beachwood, OH emal: kalambet@amersandntl.com htt://www.chromandsec.com
More informationDigital PI Controller Equations
Ver. 4, 9 th March 7 Dgtal PI Controller Equatons Probably the most common tye of controller n ndustral ower electroncs s the PI (Proortonal  Integral) controller. In feld orented motor control, PI controllers
More informationAdsorption: A gas or gases from a mixture of gases or a liquid (or liquids) from a mixture of liquids is bound physically to the surface of a solid.
Searatons n Chemcal Engneerng Searatons (gas from a mxture of gases, lquds from a mxture of lquds, solds from a soluton of solds n lquds, dssolved gases from lquds, solvents from gases artally/comletely
More informationAlgorithms for factoring
CSA E0 235: Crytograhy Arl 9,2015 Instructor: Arta Patra Algorthms for factorng Submtted by: Jay Oza, Nranjan Sngh Introducton Factorsaton of large ntegers has been a wdely studed toc manly because of
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More informationMathematical Modeling of a Lithium Ion Battery
Ecert from the Proceedngs of the COMSOL Conference 9 Boston Mathematcal Modelng of a Lthum Ion Battery Long Ca and Ralh E. Whte * Deartment of Chemcal Engneerng Unversty of South Carolna *Corresondng author:
More informationNaïve Bayes Classifier
9/8/07 MIST.6060 Busness Intellgence and Data Mnng Naïve Bayes Classfer Termnology Predctors: the attrbutes (varables) whose values are used for redcton and classfcaton. Predctors are also called nut varables,
More informationCombinational Circuit Design
Combnatonal Crcut Desgn Part I: Desgn Procedure and Examles Part II : Arthmetc Crcuts Part III : Multlexer, Decoder, Encoder, Hammng Code Combnatonal Crcuts n nuts Combnatonal Crcuts m oututs A combnatonal
More informationCHE201. I n t r o d u c t i o n t o Chemical E n g i n e e r i n g. C h a p t e r 6. Multiphase Systems
I n t r o d u c t o n t o Chemcal E n g n e e r n g CHE201 I N S T R U C T O R : D r. N a b e e l S a l m b o  G h a n d e r C h a t e r 6 Multhase Systems Introductory Statement: Phase s a state of
More informationLecture 3 Examples and Problems
Lecture 3 Examles and Problems Mechancs & thermodynamcs Equartton Frst Law of Thermodynamcs Ideal gases Isothermal and adabatc rocesses Readng: Elements Ch. 13 Lecture 3, 1 Wllam Thomson (1824 1907) a.k.a.
More information4DVAR, according to the name, is a fourdimensional variational method.
4DVaratonal Data Assmlaton (4DVar) 4DVAR, accordng to the name, s a fourdmensonal varatonal method. 4DVar s actually a drect generalzaton of 3DVar to handle observatons that are dstrbuted n tme. The
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of ExermentsI MODULE II LECTURE  GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 3.
More informationEvaluating Thermodynamic Properties in LAMMPS
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle Evaluatng hermodynamc Proertes n LAMMP Davd Keffer Deartment of Materals cence & Engneerng Unversty of ennessee Knoxvlle
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multpleregresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture  30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationSupplementary Material for Spectral Clustering based on the graph plaplacian
Sulementary Materal for Sectral Clusterng based on the grah Lalacan Thomas Bühler and Matthas Hen Saarland Unversty, Saarbrücken, Germany {tb,hen}@csunsbde May 009 Corrected verson, June 00 Abstract
More informationStatistical Evaluation of WATFLOOD
tatstcal Evaluaton of WATFLD By: Angela MacLean, Dept. of Cvl & Envronmental Engneerng, Unversty of Waterloo, n. ctober, 005 The statstcs program assocated wth WATFLD uses spl.csv fle that s produced wth
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 22 Canoncal Transformatons (Chater 9) What We Dd Last Tme Drect Condtons Q j Q j = = j P, Q, P j, P Q, P Necessary and suffcent P j P j for Canoncal Transf. = = j Q, Q, P j
More informationSome Notes on Consumer Theory
Some Notes on Consumer Theory. Introducton In ths lecture we eamne the theory of dualty n the contet of consumer theory and ts use n the measurement of the benefts of rce and other changes. Dualty s not
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationNote 10. Modeling and Simulation of Dynamic Systems
Lecture Notes of ME 475: Introducton to Mechatroncs Note 0 Modelng and Smulaton of Dynamc Systems Department of Mechancal Engneerng, Unversty Of Saskatchewan, 57 Campus Drve, Saskatoon, SK S7N 5A9, Canada
More informationPriority Queuing with Finite Buffer Size and Randomized Pushout Mechanism
ICN 00 Prorty Queung wth Fnte Buffer Sze and Randomzed Pushout Mechansm Vladmr Zaborovsy, Oleg Zayats, Vladmr Muluha Polytechncal Unversty, SantPetersburg, Russa Arl 4, 00 Content I. Introducton II.
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationThe Decibel and its Usage
The Decbel and ts Usage Consder a twostage amlfer system, as shown n Fg.. Each amlfer rodes an ncrease of the sgnal ower. Ths effect s referred to as the ower gan,, of the amlfer. Ths means that the sgnal
More informationOn New Selection Procedures for Unequal Probability Sampling
Int. J. Oen Problems Comt. Math., Vol. 4, o. 1, March 011 ISS 199866; Coyrght ICSRS Publcaton, 011 www.csrs.org On ew Selecton Procedures for Unequal Probablty Samlng Muhammad Qaser Shahbaz, Saman Shahbaz
More informationNormally, in one phase reservoir simulation we would deal with one of the following fluid systems:
TPG4160 Reservor Smulaton 2017 page 1 of 9 ONEDIMENSIONAL, ONEPHASE RESERVOIR SIMULATION Flud systems The term sngle phase apples to any system wth only one phase present n the reservor In some cases
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the faketest data; fxed
More information2 Finite difference basics
Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T
More informationIndeterminate pinjointed frames (trusses)
Indetermnate pnjonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationExperience with Automatic Generation Control (AGC) Dynamic Simulation in PSS E
Semens Industry, Inc. Power Technology Issue 113 Experence wth Automatc Generaton Control (AGC) Dynamc Smulaton n PSS E Lu Wang, Ph.D. Staff Software Engneer lu_wang@semens.com Dngguo Chen, Ph.D. Staff
More informationMODELING TRAFFIC LIGHTS IN INTERSECTION USING PETRI NETS
The 3 rd Internatonal Conference on Mathematcs and Statstcs (ICoMS3) Insttut Pertanan Bogor, Indonesa, 56 August 28 MODELING TRAFFIC LIGHTS IN INTERSECTION USING PETRI NETS 1 Deky Adzkya and 2 Subono
More informationRELIABILITY ASSESSMENT
CHAPTER Rsk Analyss n Engneerng and Economcs RELIABILITY ASSESSMENT A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng 4a CHAPMAN HALL/CRC Rsk Analyss for Engneerng Department
More informationLecture 5.8 Flux Vector Splitting
Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form
More informationProfessor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:55: Copyrght 25 Dan Dll (dan@bu.edu) Department
More informationNotforPublication Appendix to Optimal Asymptotic Least Aquares Estimation in a Singular Setup
NotforPublcaton Aendx to Otmal Asymtotc Least Aquares Estmaton n a Sngular Setu Antono Dez de los Ros Bank of Canada dezbankofcanada.ca December 214 A Proof of Proostons A.1 Proof of Prooston 1 Ts roof
More informationElectrical double layer: revisit based on boundary conditions
Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843318, USA Abstract The electrcal double layer
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng  ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng  ABCM, Curtba, Brazl, Dec. 58, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test  Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test  Wnter  Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a nonprogrammable
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationNTRU Modulo p Flaw. Anas Ibrahim, Alexander Chefranov Computer Engineering Department Eastern Mediterranean University Famagusta, North Cyprus.
Internatonal Journal for Informaton Securty Research (IJISR), Volume 6, Issue 3, Setember 016 TRU Modulo Flaw Anas Ibrahm, Alexander Chefranov Comuter Engneerng Deartment Eastern Medterranean Unversty
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationApplication of artificial intelligence in earthquake forecasting
Alcaton of artfcal ntellgence n earthquae forecastng Zhou Shengu, Wang Chengmn and Ma L Center for Analyss and Predcton of CSB, 63 Fuxng Road, Bejng 00036 P.R.Chna (emal zhou@ca.ac.cn; hone: 86 0 6827
More informationMarkov Chain Monte Carlo Lecture 6
where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways
More informationA General Class of Selection Procedures and Modified Murthy Estimator
ISS 6848403 Journal of Statstcs Volume 4, 007,. 39 A General Class of Selecton Procedures and Modfed Murthy Estmator Abdul Bast and Muhammad Qasar Shahbaz Abstract A new selecton rocedure for unequal
More informationChapter 8. Potential Energy and Conservation of Energy
Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and nonconservatve forces Mechancal Energy Conservaton of Mechancal
More informationAdiabatic Sorption of AmmoniaWater System and Depicting in ptx Diagram
Adabatc Sorpton of AmmonaWater System and Depctng n ptx Dagram J. POSPISIL, Z. SKALA Faculty of Mechancal Engneerng Brno Unversty of Technology Techncka 2, Brno 61669 CZECH REPUBLIC Abstract:  Absorpton
More informationLOGIT ANALYSIS. A.K. VASISHT Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi
LOGIT ANALYSIS A.K. VASISHT Indan Agrcultural Statstcs Research Insttute, Lbrary Avenue, New Delh0 02 amtvassht@asr.res.n. Introducton In dummy regresson varable models, t s assumed mplctly that the dependent
More informationECE559VV Project Report
ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUMRATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sumrate
More informationLecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES
COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve
More informationAirflow and Contaminant Simulation with CONTAM
Arflow and Contamnant Smulaton wth CONTAM George Walton, NIST CHAMPS Developers Workshop Syracuse Unversty June 19, 2006 Network Analogy Electrc Ppe, Duct & Ar Wre Ppe, Duct, or Openng Juncton Juncton
More information2.29 Numerical Fluid Mechanics
REVIEW Lecture 10: Sprng 2015 Lecture 11 Classfcaton of Partal Dfferental Equatons PDEs) and eamples wth fnte dfference dscretzatons Parabolc PDEs Ellptc PDEs Hyperbolc PDEs Error Types and Dscretzaton
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationHashing. Alexandra Stefan
Hashng Alexandra Stefan 1 Hash tables Tables Drect access table (or keyndex table): key => ndex Hash table: key => hash value => ndex Man components Hash functon Collson resoluton Dfferent keys mapped
More informationI + HH H N 0 M T H = UΣV H = [U 1 U 2 ] 0 0 E S. X if X 0 0 if X < 0 (X) + = = M T 1 + N 0. r p + 1
Homework 4 Problem Capacty wth CSI only at Recever: C = log det I + E )) s HH H N M T R SS = I) SVD of the Channel Matrx: H = UΣV H = [U 1 U ] [ Σr ] [V 1 V ] H Capacty wth CSI at both transmtter and
More informationLossy Compression. Compromise accuracy of reconstruction for increased compression.
Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used
More informationLesson 16: Basic Control Modes
0/8/05 Lesson 6: Basc Control Modes ET 438a Automatc Control Systems Technology lesson6et438a.tx Learnng Objectves Ater ths resentaton you wll be able to: Descrbe the common control modes used n analog
More informationComparison of the Population Variance Estimators. of 2Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 070 HIARI Ltd, www.mhkar.com Comparson of the Populaton Varance Estmators of Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data
More informationCase A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k.
THE CELLULAR METHOD In ths lecture, we ntroduce the cellular method as an approach to ncdence geometry theorems lke the SzemerédTrotter theorem. The method was ntroduced n the paper Combnatoral complexty
More informationDependent variable for case i with variance σ 2 g i. Number of distinct cases. Number of independent variables
REGRESSION Notaton Ths rocedure erforms multle lnear regresson wth fve methods for entry and removal of varables. It also rovdes extensve analyss of resdual and nfluental cases. Caseweght (CASEWEIGHT)
More informationIrregular vibrations in multimass discretecontinuous systems torsionally deformed
(2) 4 48 Irregular vbratons n multmass dscretecontnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscretecontnuous systems consstng of an arbtrary number rgd bodes connected
More informationExperiment 1 Mass, volume and density
Experment 1 Mass, volume and densty Purpose 1. Famlarze wth basc measurement tools such as verner calper, mcrometer, and laboratory balance. 2. Learn how to use the concepts of sgnfcant fgures, expermental
More informationMETHOD OF NETWORK RELIABILITY ANALYSIS BASED ON ACCURACY CHARACTERISTICS
METHOD OF NETWOK ELIABILITY ANALYI BAED ON ACCUACY CHAACTEITIC ławomr Łapńsk hd tudent Faculty of Geodesy and Cartography Warsaw Unversty of Technology ABTACT Measurements of structures must be precse
More informationStatistical Energy Analysis for High Frequency Acoustic Analysis with LSDYNA
14 th Internatonal Users Conference Sesson: ALEFSI Statstcal Energy Analyss for Hgh Frequency Acoustc Analyss wth Zhe Cu 1, Yun Huang 1, Mhamed Soul 2, Tayeb Zeguar 3 1 Lvermore Software Technology Corporaton
More informationNON LINEAR ANALYSIS OF STRUCTURES ACCORDING TO NEW EUROPEAN DESIGN CODE
October 117, 008, Bejng, Chna NON LINEAR ANALYSIS OF SRUCURES ACCORDING O NEW EUROPEAN DESIGN CODE D. Mestrovc 1, D. Czmar and M. Pende 3 1 Professor, Dept. of Structural Engneerng, Faculty of Cvl Engneerng,
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationAssignment 5. Simulation for Logistics. Monti, N.E. Yunita, T.
Assgnment 5 Smulaton for Logstcs Mont, N.E. Yunta, T. November 26, 2007 1. Smulaton Desgn The frst objectve of ths assgnment s to derve a 90% twosded Confdence Interval (CI) for the average watng tme
More informationThe optimal delay of the second test is therefore approximately 210 hours earlier than =2.
THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 615086 provdes approxmaton formulas for the PF for smple
More informationTCP NewReno Throughput in the Presence of Correlated Losses: The SlowbutSteady Variant
TCP NewReno Throughut n the Presence of Correlated Losses: The SlowbutSteady Varant Roman Dunaytsev, Yevgen Koucheryavy, Jarmo Harju Insttute of Communcatons Engneerng Tamere Unversty of Technology Tamere,
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationin a horizontal wellbore in a heavy oil reservoir
498 n a horzontal wellbore n a heavy ol reservor L Mngzhong, Wang Ypng and Wang Weyang Abstract: A novel model for dynamc temperature dstrbuton n heavy ol reservors s derved from and axal dfference equatons
More informationPulse Coded Modulation
Pulse Coded Modulaton PCM (Pulse Coded Modulaton) s a voce codng technque defned by the ITUT G.711 standard and t s used n dgtal telephony to encode the voce sgnal. The frst step n the analog to dgtal
More informationJAB Chain. Longtail claims development. ASTIN  September 2005 B.Verdier A. Klinger
JAB Chan Longtal clams development ASTIN  September 2005 B.Verder A. Klnger Outlne Chan Ladder : comments A frst soluton: Munch Chan Ladder JAB Chan Chan Ladder: Comments Black lne: average pad to ncurred
More informationANOVA. The Observations y ij
ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2
More informationGravitation and Spatial Interaction
Unversty College London Lectures on Urban Modellng January 2017 Gravtaton and Satal Interacton Mchael Batty m.batty@ucl.ac.u @mchaelbatty 16 January, 2017 htt://www.satalcomlexty.nfo/ Lectures on Urban
More informationTopology optimization of plate structures subject to initial excitations for minimum dynamic performance index
th World Congress on Structural and Multdsclnary Otmsaton 7 th 2 th, June 25, Sydney Australa oology otmzaton of late structures subject to ntal exctatons for mnmum dynamc erformance ndex Kun Yan, Gengdong
More informationLaboratory 3: Method of Least Squares
Laboratory 3: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly they are correlated wth
More informationMA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials
MA 323 Geometrc Modellng Course Notes: Day 13 Bezer Curves & Bernsten Polynomals Davd L. Fnn Over the past few days, we have looked at de Casteljau s algorthm for generatng a polynomal curve, and we have
More informationASLevel Maths: Statistics 1 for Edexcel
1 of 6 ASLevel Maths: Statstcs 1 for Edecel S1. Calculatng means and standard devatons Ths con ndcates the slde contans actvtes created n Flash. These actvtes are not edtable. For more detaled nstructons,
More informationTranslational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.
Lesson 12: Equatons o Moton Newton s Laws Frst Law: A artcle remans at rest or contnues to move n a straght lne wth constant seed there s no orce actng on t Second Law: The acceleraton o a artcle s roortonal
More informationLecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The leastsquares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationTopic 23  Randomized Complete Block Designs (RCBD)
Topc 3 ANOVA (III) 31 Topc 3  Randomzed Complete Block Desgns (RCBD) Defn: A Randomzed Complete Block Desgn s a varant of the completely randomzed desgn (CRD) that we recently learned. In ths desgn,
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More information36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to
ChE Lecture Notes  D. Keer, 5/9/98 Lecture 6,7,8  Rootndng n systems o equatons (A) Theory (B) Problems (C) MATLAB Applcatons Tet: Supplementary notes rom Instructor 6. Why s t mportant to be able to
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationABSTRACT. 1. Introduction. propagation of. with respect to. Method. dered in. terms of the velocity Cartesin. waves in the. the Atmosphere. D one.
Journal of Aled Mathematcs and Physcs,, 13, 1, 117 htt://dx.do.org/1.436/jam..13.143 Publshed Onlne October 13 (htt://www.scr.org/journal/jam) Numercal Smulaton of AcoustcGravty Waves Proagaton n a Heterogeneous
More informationFormulation of Circuit Equations
ECE 570 Sesson 2 IC 752E Computer Aded Engneerng for Integrated Crcuts Formulaton of Crcut Equatons Bascs of crcut modelng 1. Notaton 2. Crcut elements 3. Krchoff laws 4. ableau formulaton 5. Modfed nodal
More informationSimulation and Random Number Generation
Smulaton and Random Number Generaton Summary Dscrete Tme vs Dscrete Event Smulaton Random number generaton Generatng a random sequence Generatng random varates from a Unform dstrbuton Testng the qualty
More informationIII. Econometric Methodology Regression Analysis
Page Econ07 Appled Econometrcs Topc : An Overvew of Regresson Analyss (Studenmund, Chapter ) I. The Nature and Scope of Econometrcs. Lot s of defntons of econometrcs. Nobel Prze Commttee Paul Samuelson,
More informationChapter 3. Estimation of Earthquake Load Effects
Chapter 3. Estmaton of Earthquake Load Effects 3.1 Introducton Sesmc acton on chmneys forms an addtonal source of natural loads on the chmney. Sesmc acton or the earthquake s a short and strong upheaval
More information