Északkele-Magyarország, Gömör-Tornai-Karsz, Aggeleki Nemzei Park A Világ Kulurális és Természei Örökségének része, UNESCO (1995) VITUKI JósvafJ svafıi i Papp Ferenc Karszvízkua zkuaó Állomás: Karszforrások sok vízhozam v válozv lozásainak vizsgálaa 1959-ıl Vizsgál források (1959-1993): Jósva-forrás (ereei forrásszáj) Jósva-forrás (alsóbarlangi forrásszáj) Komlós-forrás Nagy-Tohonya-forrás Kis-Tohonya-forrás Lófej-forrás Babokú-forrás Szabókú-forrás A Nagyohonya forrás s vízhozam v iısora 1964 jan. 1-ıl 1 l 1992 ec. 31-ig noh 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 Nagyohonya forrás A karszos járarenszerek hierarchiája: 1. barlangi fıág 2. barlangi mellékágak 3. fıörésrenszer 4. ömbön belüli mellékörések 5. ömbön belüli mikrorepeések 0 2000 4000 6000 8000 10000 Karszmoell (Maucha László): A kiürülési vízhozam iısor logarimikus lépékben ábrázoljuk: Apaási vízhozam iısorok maemaikai vizsgálaa: napi álagos vízhozam iısorok vizsgálaa 20-30 éves aasor 5 forrásra: Jósva-forrás, Komlós-forrás, Nagy- Tohonya- forrás, Kis-Tohonya-forrás, Lófej-forrás vízhozam logarimusok érékekre regressziós egyenesek illeszése 30000 25000 1977. áprilisól 1978. májusól 1979. áprilisól 1983. márciusól 1989. májusól vízhozam (m 3 /nap) 20000 15000 10000 A nagyobb szélességő járaok visszauzzaszják a kisebbeke. Az ö, nagyságrenileg különbözı álag kereszmeszeő ároló egymás uán ürül le. 5000 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 nap Kis-Tohonya-forrás kiválaszo apaási szakaszai
mereekség és szórásnégyze ábrázolása 3 ároló renszer meghaározása Regressziós egyenesek illeszésének mószerével meghaározo Empirikus úon meghaározo Napok mereekség napok 1. ároló 1-5 -0,24 1-4 Auocorrelaions 0.0 0.2 0.4 0.6 0.8 1.0 Series : nohlsr Kis-Tohonya-forrás 2. ároló 6-17 -0,10 5-23 3. ároló 18-33 -0,06 24-56 1. ároló 1-2 -0,57 1-2 400 500 Lag Komlós-forrás 2. ároló 3-11 -0,12 3-12 Jósva-forrás Lófej-forrás 3. ároló 12-42 -0,06 13-43 1. ároló 1-3 -0,95 1-2 2. ároló 4-5 -0,24 3-6 3. ároló 6-16 -0,03 7-18 1. ároló 1-4 -0,30 1-5 2. ároló 5-13 -0,13 6-17 3. ároló 14-37 -0,07 18-61 pernoh 0 10 20 30 40 Nagy-Tohonya-forrás 1. ároló 1-10 -0,12 1-11 2. ároló 11-18 -0,07 12-31 3. ároló 19-57 -0,05 32-78 0 1000 2000 3000 4000 5000 Non-smoohe perioogram Smoohe perioogram Yearly average an smoohe ren Empirical an LOESS smoohe yearly variance Yearly average of Deseasonalise resiuals (10^(iosorspec$spec/10))[1:150] 0 5*10^7 10^8 1.5*10^8 0.0 0.001 0.002 0.003 0.004 iosorspec$fre[1:150] nameves[, 3] 200 400 600 sr(nlrcikl[, 4]) 200 300 400 500 nlsreves[, 3] -0.15-0.05 0.05 0.15 (10^(iosorspec$spec/10))[1:150] 0 5*10^6 1.5*10^7 nleves[, 3] -60-20 0 20 40 60 nlsreves[, 4] 0.5 1.0 1.5 2.0 logrsar1[2:len] -2 0 2 4 6 0.0 0.001 0.002 0.003 0.004 iosorspec$fre[1:150] 2 4 6 8 10 logio, Hurs=0.6603, s.err=0.0009 Perioogram of erene series Derene resiuals Yearly variance of resiuals Classic R/S Deszezonalizál l vízhozamok, v Nagyohonya forrás Bi-monhly isribuions Deszezonalizál vízhozamok, Nagyohonya forrás nohlsr 0 2 4 6 8 10 0 2000 4000 6000 8000 10000
Long memory processes Shor an long memory - auocorrelaions ecay a exponenial versus hyperbolic rae Euivalen for long memory - specral ensiy has pole a zero, meaning i ens o infiniy a zero a polynomial spee. Reason for euivalence - a Tauber ype heorem Inicaors of long memory Nonparameric saisics Rescale ajuse range or R/S Variance plo Log-perioogram (Geweke-Porer Huak) Derene flucuaion analysis Higuchi meho Parameric saisics Unersaning R/S Hurs s original iea: how o plan he capaciy of he reservoir as o prouce a uniform runoff (ouflow). Accumulae he regisere inflow up o year n Subrac from i he ieal uniform ouflow he maximum shows he greaes surplus, he minimum he greaes shorage, herefore The range (he ifference of he above wo) gives he neee capaciy of he reservoir Ge ri of scaling: sanarize he range - R/S Growh of R/S For saionary processes R/S ens o infiniy, bu is spee is characerisic R/S is proporional o n H ; H is calle he Hurs exponen On he log-log scale: he slope of he fie line Roughly: For shor memory processes H=1/2, for long memory ones 1/2<H<1 Convenional R/S: biase, lack of isribuion heory, no es or confience inervals, sensiive for shor mem componen if presen. Lo s moificaion - convergence o Brownian Brige. Tes of long memory; ofen acceps shor memory when long is presen, bu reliably rejecs long mem when only shor is presen. Tau s graphical R/S; ge ou he mos of he aa: rop blocks from he beginnig, reesimae R/S from he shorer aa, log-log plo all ogeher an fi a sraigh line (regression) Convenional R/S logrsar1[2:len] -2 0 2 4 6 2 4 6 8 10 logio, Hurs=0.6603, s.err=0.0009
5 6 7 8 9 10 Graphical R/S Saisics base on parameric moels RperS[1:(szamol - 2), 1] 3 4 5 6 7 Fracional ARIMA moel fracional ifferencing : Replace x in he Taylor expansion of he funcion (1-x) wih he backshif operaor, Fi an ARMA moel o he fracionally ifference process is connece o he Hurs exponen by =H-1/2 Esimae H using a While-ype esimaor base on he perioogram R/S: 0.6807567 Linear long-memory moel : fracional ARIMA process (Monanari e al., Lago Maggiore, 1997) Fracional ARIMA-moel (wih skewe, inep. value noise): Φ ( (1 X =Ψ( ε Fiing is one by he While-esimaor: uie robus; consisen an asympoically normal for linear processes riven by innovaons wih finie fourh momens (Giraiis an Surgailis, 1990) 2 0.34 (1 0.80 B+ 0.12 B ) (1 X = (1+ 0.21 ε H=0.846 (sanar error: 0.014) Beran s p-value: 0.558 (inicaes gooness of fi) Innovaions can be reconsruce using he inverse of he filer above Comparison of auocorrelaions Original (black) an simulae FARIMA (2,,1) (re) wih inepenen-value innovaions Mulifracal Propery The basic probabilisic characerisics shoul remain unchange (monofracal) or similar (mulifracal) wih he change of scale Momens, cumulans, an uaniles are cosiere Change of scale means aggregaion The concep of fracals, ha i is similariy on all scales, eners sochasic analysis by he efiniion of Self-Similar processes wih Saionary Incremens (sssi). The sochasic process Y() is calle sssi, if i has saionary incremens an, for all c>0 real numbers, H Y ( c) = c Y ( ) where = means he eualiy of finie-imensional isribuions. The parameer H is referre o as he Hurs parameer or Hurs exponen.
Simple fracal processes Wiely use examples of such processes are Fracional Brownian Moions (FBM), where he parameer H (0;1), or Lévy walks. As a sraighforwar conseuence of he self-similariy euaion he -h absolue momens M of Y() saisfy H K ( ) M ( ) = c( ) = c( ) i.e. he logarihm of M () scales linearly wih log() an he coefficien is H, meaning he -specrum o be a linear funcion of, in oher wors hese processes are monofracals. Mulifracaliy Many empirical suies foun, however, ha he - specrum of flucuaions is a non-linear convex funcion, ha is he Hurs exponen also epens on, i.e.: -K() = H(). A wier class of mulifracal processes preserves scale invariance properies in cerain respec. These properies are generally characerise by he exponens or -specrum K() ha govern he power law scaling of he absolue momens of is flucuaions. Momen-mulifracal process More formally, a sochasic process is calle mulifracal, if K ( ) M (, h) ( c( h) ) where M(,h) sans for he -h absolue momens of he sochasic process wih (rescale) saionary incremens ( ) M (, ) = E Y ( s+ ) Y ( s) = E( Y ( ) Y (0) ) an K() is a non-linear convex funcion. Cumulan-mulifracal process Suppose ha he m h orer cumulan, cum m (Y ()), exiss. A sochasic process Y() is calle mulifracal in he cumulan sense or c-mulifracal for shor, if, for all posiive inegers m, he logarihm of cum m (Y()) scales linearly wih log() an he coefficien is m H(m), i. e. log( cum m (Y()) ) = m H(m) log() + c(m) Noe ha he consans c(m) in he above efiniion may ake he value -, if he corresponing cumulans are zero. So far as monofracals are concerne, he wo efiniions, i.e. momen an cumulan mulifracaliy, are euivalen. Evience for Mulifracal Propery, Momens Nameny loess_sres Zahony rig8resi Evience for Mulifracal Propery Cumulans 30 Felfele: m=2,3,4,5,6 renu kumulansok log of momens -4-3 -2-1 0 1 log of momens 0 5 10 15 25 20 5 10 15 20 25 log of egree of aggregaion Nameny loess_sres 5 10 15 20 25 log of egree of aggregaion Zahony rig8res 15 K() 0.0 0.5 1.0 K() 0.0 0.2 0.4 0.6 0.8 1.0 10 5 1 2 3 4 5 6 egree of momens 1 2 3 4 5 6 egree of momens 0 1 1.5 2 2.5 3 3.5 4 4.5