Almost power law : Tempered power-law models (T-FADE)

Transcription

1 Almos powr law : Tmprd powr-law modls T-FADE Yong Zhang Dsr Rsarch Insiu Novmbr 4, 29

2 Acknowldgmns Boris Baumr Mark Mrschar Donald Rvs

3 Oulin Par Spac T-FADE modl. Inroducion 2. Numrical soluion 3. Momn analysis 4. Th spac T-FADE wih im nonlocaliy 5. Applicaions -d & Muli-scaling T-FADE Par 2 Tim T-FADE modl. Why? Wha? 2. Paramr simaion

4 Dnsiy. Inroducion Why do w nd Tmprd Sabl Jumps? =. -6 X2 Sandard Lévy moion:. Arbirarily larg jumps 2. Divrgn spaial momns Mangna & Sanly [994]; hchkin, Klafr & Mzlr [23]; Sokolov al. [24]; Rosiński [27]; ara & Dl-asillo-Ngr [27]

5 Sabl jumps vs. Tmprd sabl jumps Sabl jumps Tmprd SJ Lévy masur. Inroducion φ d φ d Dnsiy Fracionaldrivaiv Momns Larg jumps f, f, f, = Infini Fini? Powr-law Eponnial d

6 A spaially aprd fade T-FADE Modl D V = Th sandard fade Th govrning quaion. Inroducion [ ] [ ] [ ] = = D D V D V

7 . Inroducion TRW and is scaling limi Jump Siz Varianc Fini Infini Waiing Tim Momns Infini Fini Sub-diffusion Tim fade Sub/Supr-diffusion SpacTim fade Brownian Moions ADE Tmprd supr-diffusion Spac T-FADE Supr-diffusion Spac fade

8 Oulin Par Spac T-FADE modl. Inroducion 2. Numrical soluion 3. Momn analysis 4. Th spac T-FADE wih im nonlocaliy 5. Applicaions -d & Muli-scaling T-FADE Par 2 Tim T-FADE modl. Why? Wha? 2. Paramr simaion

9 Lagrangian approimaion Moion procss [ ] j j j d d D f V dx ξ τ =,, j dξ can b gnrad by h ponnial rjcion mhod [Baumr & Mrschar, 29] 2. Numrical soluion =, i j n j j i f h Grünwald approimaion Implici Eulrian fini diffrnc soluion [ ] [ ] = D D V

10 2. Numrical soluion -2 =.3, various Normalizd oncnraion = E-5 = E-4 = E-3 = E-2 = 5E-2 = E- Gaussian Sandard fade Disanc from sourc Gaussian T-FADE: Inrmdia/ransin anomalous diffusion

11 Sampl pah of jump sizs - 2. Numrical soluion Effc of runcaion paramr for < < 2 = -6 = -2 Sampl pah of jump sizs Numbr of jumps Numbr of jumps Snapsho, = - > / = -2 = = -3

12 Oulin Par Spac T-FADE modl. Inroducion 2. Numrical soluion 3. Momn analysis 4. Th spac T-FADE wih im nonlocaliy 5. Applicaions -d & Muli-scaling T-FADE Par 2 Tim T-FADE modl. Why? Wha? 2. Paramr simaion

13 T-FADE [ ] = D V = M 2 / 2 / 2 / ] [ 2 = D S V E = D 2 2 = σ Mass Man Varianc Skwnss 3. Momn analysis D V = 2 2 onvrg o ross-ovr im τ ~ D

14 3. Momn analysis Numrical ampls: Spaial momns for h spac-only T-FADE modl Spaial Momns Kurosis Skwnss Man Mass Varianc Tim Skwnss and Kurosis Kurosis =.97 Skwnss =.97 Skwnss =.97 Kurosis = Tim

15 Oulin Par Spac T-FADE modl. Inroducion 2. Numrical soluion 3. Momn analysis 4. Th spac T-FADE wih im nonlocaliy 5. Applicaions -d & Muli-scaling T-FADE Par 2 Tim T-FADE modl. Why? Wha? 2. Paramr simaion

16 [ ] δ β β Γ = D V T T T T T Th T-FADE Modl δ β β Γ = D V T T T [ ] = D V M M M M M β β D V M M M = Th fracal mobil/immobil modl [Schumr al., 23] 4. Th spac T-FADE wih im nonlocaliy

17 4. Th spac T-FADE wih im nonlocaliy Normalizd oncnraion Normalizd oncnraion a T= =., β =.5, =.5, = b T=5 Evoluion of h plum FD RW o m im Disanc Disanc 3 Normalizd oncnraion Normalizd oncnraion c Linar-Linar plo of a Mobil Immobil Toal phas Disanc 3 d Linar-Linar plo of b Disanc 3

18 [ ] = D V m m m m m β 2 / β Γ M m 2 / β Γ Γ V E m β β σ f D f V m V E m D m 2 2 σ Mass Man Varianc Early im La im / β Γ M m Momn analysis 4. Th spac T-FADE wih im nonlocaliy

19 Oulin Par Spac T-FADE modl. Inroducion 2. Numrical soluion 3. Momn analysis 4. Th spac T-FADE wih im nonlocaliy 5. Applicaions -d & Muli-scaling T-FADE Par 2 Tim T-FADE modl. Why? Wha? 2. Paramr simaion

20 Mass Fracion [dimnsionlss] 5. Applicaions Applicaion #: MADE Bromid Spaial momns Mass Dcay 2 Man 4 3 Varianc 2 Displacmn Man Displacmn [m] Varianc [m 2 /day] Tim [day] 4 6 Tim [day] 4 6 Tim [day] Skwnss [dimnsionlss] 4 Skwnss 5 Kurosis Kurosis [dimnsionlss] Tim [day] Tim [day]

21 5. Applicaions - Plum in discr fracur nworks 5 Masurmn FADE T-FADE 3 = Disanc m Bs-fi 3 =2 Disanc m - Prdicion Disanc m 2 4 Disanc m 2 4 Disanc m

22 5. Applicaions Fracurs Mon arlo T-FADE a 2 b c =.6, =. M45 =.5 Y m 2 X m Y m - 2 X m Y m - 2=.6, 2 =. M-45 =.5 2 X m d 2 f =.8, =.6 M45 =.5 Y m Y m Y m 2=.8, 2 =.6 M-45 = X m - 2 X m - 2 X m

23 Par 2 Tim T-FADE. Why? Wha? 2. How o prdic modl paramrs?

24 Why do w nd h im T-FADE? Brakhrough urv - BT fade -5-6 ADE Tim

25 Sabl waiing ims vs. Tmprd SWT Lévy masur Dnsiy Fracionaldrivaiv Momns displacmn Larg WTs Sabl WT Tmprd SWT φ d φ d g, g, g,, Fini Powr-law d Fini ADE Eponnial

26 Th im T-FADE wih a advcion-disprsion opraor Th prvious Tim fade Tim T-FADE M M M M L = β β δ β β Γ = m L T T M M L = β g m L T T T T δ β β β =

27 Tim T-FADE Scaling limi of TRW Jump Siz Varianc Fini Infini Waiing Tim Momns Infini Fini Sub-diffusion Tim fade Sub/Supr-diffusion SpacTim fade Tim T-FADE Brownian Moions ADE Spac T-FADE Supr-diffusion Spac fade

28 Lagrangian approimaion of sampl pah Sampl pah of waiing ims Sampl pah of waiing ims Numbr of wai =.5, β =.5 =.5, β = Numbr of wai for < < T = -6 T = -2

29 Influnc of paramrs Normalizd oncnraion T = -5 T = -3 Toal phas ~ - Mobil phas m ~ Tim Normalizd oncnraion Tim Normalizd oncnraion T = - T = Normalizd oncnraion Tim Tim

30 Influnc of paramrs Normalizd oncnraion =..5 Scal ind conrols h powr-law slop Tim Normalizd oncnraion β =. apaciy cofficin β conrols h mass raio Tim

31 Applicaion [Mrschar al., 28] Dy concnraion ppb L=34.35 km km km ADE Masurd Tim sinc injcion hour Fid Missouri Rivr dy BT T =. capurs havy la-im ail in h BT oncnraion [M/L 3 ] T= Disanc from sourc [L] Tracr snapsho a porscal mdium T =. capurs rnion nar h sourc

32 Par 2 Tim T-FADE. Why? Wha? 2. How o prdic modl paramrs?

33 m Paramr simaion 4.5 y m 97 z m 45 Dbris Flow Floodplain Lv hannl

34 Z m Xm Xm.6..4 Immobil.8.2 Mobil..6 Immobil.8 Mobil oncnraion oncnraion Z m Tim yar Tim yar 3 4

35 onclusion ADE T-FADE FADE

36 Rfrncs. Mrschar, M. M., Y. Zhang, and B. Baumr. Tmprd anomalous diffusion in hrognous sysms, Gophysical Rsarch Lrs, 35, L743, Baumr, B., and M. M. Mrschar. Tmprd sabl Lévy moion and ransin supr-diffusion, Journal of ompuaional and Applid Mahmaics, 29, in prss. 3. Zhang, Y., B. Baumr, and D. M. Rvs. A spac-aprd vcor Lévy moion modl o simula prasympoic ranspor in rgional-scal fracurd mdia, submid, 29.