Multivariate Auto-Regressive Model for Groundwater Flow Around Dam Site

Transcription

1 Mulivariae uo-regressive Model for Groundwaer Flow round Da Sie Yoshiada Mio ), Shinya Yaaoo ), akashi Kodaa ) and oshifui Masuoka ) ) Dep. of Earh Resources Engineering, Kyoo Universiy, Kyoo, 66-85, Japan. bsrac: ie series analysis is carried ou using ulivariae auo-regressive odel for he onioring of eporal variaions of groundwaer behavior around he fill da sie. he filered ie series referring aospheric eperaure is suggesed as he ain anipulaed variables in order o ake he eling snow, which causes a sudden rise in he level of he river, ino accoun. Is effeciveness is verified hrough analyzing he power conribuion. he resuls of he regression show ha he ulivariae auo-regressive odel using he proposed variables is very effecive ool for he esiaion of he groundwaer behavior of he da sie.. Inroducion Monioring of eporal variaions of groundwaer flow behavior around he copleed da sie is indispensable o secure he peranen sabiliy of da and is foundaion. he observed daa for da onioring, which varies in he ie doain, is coposed of ulifarious daa such as groundwaer pressure, da and foundaion displaceen, flow rae of he river, weaher condiion and so on. In general, he sabiliy of da and foundaion is saisically exained, day-by-day, hour-by-hour or soeies inue-by-inue, referring he differences beween he observed daa and he prediced daa using ulivariae regression odel. owever, he regression resuls do no always have a desirable accuracy, since he observed daa ofen includes any of he effec of he pas behavior, which is no aken ino consideraion in he case of ulivariae regression odel. herefore, i is essenial o carry ou ulivariae ie series analysis in order o reflec he effec of he pas daa. In his sudy, ulivariae auo-regressive odel (vecor auo-regressive odel), which is a variaion of he ie series analysis odel, is applied o he regression of eporal variaions of groundwaer flow head around a da sie, and he facor analysis of variaions is carried ou by evaluaing he relaive power conribuion.. Mulivariae auo-regressive odel In he case ha sochasic process can be expressed as x ( ) a( ) x( ) u( ) = M = x() is called as he M-h order auo-regressive process where, u(), and a() denoe ie, whie noise and auo-regressive coefficien, respecively. Exending his process o k diensional ulivariae case, we can obain a ulivariae auo-regressive process;

2 ( ) ( ) x( ) u( ) x = M = where u() and () denoe whie noise vecor (k diensional) and auo-regressive coefficien arix (k x k diensional), respecively. 3. Relaive power conribuion ssuing ha here is no correlaion beween whie noises of differen variables, he power specru is given by p ii k ( f ) = ( f ) j= ij σ jj where σ i j = ( f ) ( f ) ( f ) M ( f ) = ( ) exp( πif) ( f ) = p = M pk ( f ) L p ( f ) O k f L pkk f his expression iplies power specru can be divided ino effecs of p noise sources. herefore we can obain he relaive conribuion of noise o he variaion of x i () which is relaed o frequency f, relaive power conribuion r ij ( f ) ( ( f ) ) ij p ( f ) ii M jj σ =. his value allows us o carry ou facor analysis of ie series daa. 4. ie series daa of he da sie his sudy is applied o a rock fill da, which is locaed in norhern disric of Japan. he heigh, he cres lengh, and he volue of he da are 63.5, 6, and,387 3, respecively. he da foundaion is coposed of ypes of Quaernary pyroclasic rocks spewed ou in differen sages. Lower par of he sequence is fresh and highly pereable welded uff; because he less pereable, poorly welded zone has been eroded and only he highly welded zone reains where open, super conducive colunar joins are disribued. On he oher hand, upper par of he sequence shows lower pereabiliy. he da sie, lower pereable par is disribued on

3 he righ bank of he river, on he oher hand, he upper ipereable par exiss on he lef. boreholes for observing groundwaer head (waer pressure) are esablished around he da sie as shown in Figure. Mos of he are locaed on he down-sreaside of he grou line. ie series daa of he groundwaer head was auoaically observed wihin a given period of ie (fro Sepeber 997 o June 999, occasionally, o Deceber 998) in order o grasp he groundwaer flow behavior around he da sie spaially and eporally. he waer level of da reservoir, he rainfall and he air eperaure were also observed as ie elapsed. eporal variaions of he observed daa ( dau/day) are shown in Figure. he ie series daa of he groundwaer head as he conrolled variables (arge variables) and he oher ies series are reaed as anipulaed variables (explanaory variables). Figure Layou of he boreholes /7/ 997/9/ 997// 998// 998/3/ 998/5/ 998/7/ 998/9/ 998// 999// 999/3/ 999/5/ /7/ 997/9/ 997// 998// 998/3/ 998/5/ 998/7/ 998/9/ 998// 999// 999/3/ 999/5/ Dae Dae (a) Groundwaer head Figure () eporal daa observed a he da sie (b) Waer level of he reservoir

4 /7/ 997/9/ 997// 998// 998/3/ 998/5/ 998/7/ 998/9/ 998// 999// 999/3/ 999/5/ 997/7/ 997/9/ 997// 998// 998/3/ 998/5/ 998/7/ 998/9/ 998// 999// 999/3/ 999/5/ Dae Dae (c) Rainfall (d) ir eperaure Figure () eporal daa observed a he da sie 5. Filering of air eperaure daa Since he sudy da sie is locaed in he norhern par of Japan, a vas snow lays upsrea area of he da sie in every winer season and hey els in every spring season. his aospheric cycle causes a sudden rise in he waer level of he river every spring and raises he groundwaer head rapidly. owever here is no anipulaed variable o be able o explain such a sudden change. hus, air eperaure daa, which sees o have possibiliy o explain he snow-eling phenoenon, is filered wih F ( ) ( ) π = ax sin, ( ) = for he period fro March o ugus F for he period fro Sepeber o February where () is he air eperaure as he funcion of ie, and is he ean air eperaure fro he end of he March o he beginning of ugus. ha filering funcion considers he fac ha he air eperaure in he da sie exceeds degree cenigrade in he end of he March and he snow coninues o lay unil in he beginning of he ugus. Figure 3 shows he filered ie series of he air eperaure /7/ 997/9/ 997// 998// 998/3/ 998/5/ 998/7/ Dae 998/9/ 998// 999// 999/3/ 999/5/ Figure 3 eporal variaion of F()

5 6. Regression ypes of regressions are carried ou in his sudy. Firsly, a ulivariae auo-regressive odel is defined as R R R u =... where (): he groundwaer flow head, (): he waer level of he reservoir, R(): he rainfall, (): he air eperaure, (): coefficien arix, and u(): whie noise vecor. Secondly, he odel using F() is defined as F R F R F R u =.... Figure 4 shows illusraions of he regression resuls. he finess of he regression curve of he odel wih F() is beer han ha wihou F(). 7. Facor analysis Relaive power conribuion of each easuring poin is calculaed as shown in Figure 5. able shows he qualiaive caegorical esiaions of conribuions and regression accuracy of each easuring poin. he conribuion of iself (auo-correlaion) is higher a every easuring poin. he poins where he conribuion of he waer level of he reservoir is rearkable are locaed on he righ bank, which is ore pereable. he poins where he conribuion of he rainfall or he eling snow is rearkable are locaed in cerain regions. he poins where he noable conribuion of F() is perceived are locaed a he ridge of he lower foraion of pyroclasic rocks as shown in Figure 6. Such geo-orphologically characerized poins are generally ap o be unsauraed in he dry season (non eling-snow season) because of highly pereable hydraulic propery and opological disadvanage o be culivaed. herefore he hydraulic conduciviy of hose pars varies annually considering pereabiliy change under sauraed/unsauraed condiion, and he saionariy of he ie series of hose pars seeed o be los. hus he conribuion of F() would be rearkable a hose poins.

6 Observaion wihou F() wih F() /7/ 997/9/ 997// 998// 998/3/ 998/5/ 998/7/ 998/9/ 998// 999// 999/3/ 999/5/ Dae (a) GW9 Observaion wihou F() wih F() /7/ 997/9/ 997// 998// 998/3/ 998/5/ 998/7/ 998/9/ 998// 999// 999/3/ 999/5/ Dae Figure 4 (b) L4 Illusraions of he regression using Mulivariae auo-regressive odel

7 ower specral densiy 5 5 ower specral densiy F() ir eperaure Rainfall Waer level of he reservoir Groundwaer head Frequency (cycle/day) Frequency (cycle/day) (a) GW ower specral densiy 5 5 ower specral densiy F() ir eperaure Rainfall Waer level of he reservoir Groundwaer head Frequency (cycle/day) Frequency (cycle/day) Figure 5 Illusraions of he relaive power conribuion (b) GW able. Qualiaive esiaion of relaive power conribuion and regression accuracy oin () R() () F() () ccuracy GW5 GW9 GW GW GW GW3 GW4 GW5 NO4 NO5 L L4 L6 L7 L8 L9 L R3 R4 R5

8 EL[] 65 6 Upper foraion he conribuion of F() is rearkable. 55 Lower foraion he conribuion of F() is no rearkable L L L6 L4 L7 L8 L9 GW4 GW5 (a) - secion EL[] B B Upper foraion he conribuion of F() is rearkable. Lower foraion he conribuion of F() is no rearkable GW5 GW GW GW9 R5 GW5 R4 R3 GW5 Figure 6 (b) B-B secion Geological cross secion of he da sie 8. Conclusion In his sudy, ie series analysis is carried ou using ulivariae auo-regressive odel for he onioring of eporal variaions of he groundwaer behavior around he fill da sie. he proposed filered ie series referring air eperaure is verified o be effecive as one of he anipulaed variables in order o ake he eling snow ino accoun. s he resuls of exainaions, ulivariae auo-regressive odel using he proposed variables is very effecive ool for he esiaion of eporal groundwaer behavior of he da sie, and relaive power conribuion is also effecive o grasp he facor of he echanis of eporal variaion. References []. kaike : Soe probles in he applicaion of he cross-specral ehod, Specral nalysis of ie Series, pp.8-7, 969. []. kaike: On he use of linear odel for he idenificaion of feedback syses, nn. Ins. Sais. Mah., Vol., pp , 968.