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❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s✿ ❆♥ ❡♠♣✐r✐❝❛❧ ❛♥❛❧②s✐s ♦❢ t❤❡ ❧♦♥❣✲r✉♥ ❛♥❞ s❤♦rt✲r✉♥ ❞②♥❛♠✐❝s ❢r♦♠ t❤❡ ●✲✼ ❈♦✉♥tr✐❡s ❈❛t❤② ❨✐✲❍s✉❛♥ ❈❤❡♥ ❚❤♦♠❛s ❈✳ ❈❤✐❛♥❣ ❲♦❧❢❣❛♥❣ ❑❛r❧ ❍är❞❧❡ ❈❤✉♥❣✲❍✉❛ ❯♥✐✈❡rs✐t② ▲❡❇♦✇ ❈♦❧❧❡❣❡ ♦❢ ❇✉s✐♥❡ss✱ ❉r❡①❡❧ ❯♥✐✈❡rs✐t② ▲❛❞✐s❧❛✉s ✈♦♥ ❇♦rt❦✐❡✇✐❝③ ❈❤❛✐r ♦❢ ❙t❛t✐st✐❝s ❈✳❆✳❙✳❊✳ ✕ ❈❡♥t❡r ❢♦r ❆♣♣❧✐❡❞ ❙t❛t✐st✐❝s ❛♥❞ ❊❝♦♥♦♠✐❝s ❋✐❣✉r❡s✴❈❍❯❧♦❣♦✳♣♥❣ ❍✉♠❜♦❧❞t✕❯♥✐✈❡rs✐tät ③✉ ❇❡r❧✐♥ ❤tt♣✿✴✴❧✈❜✳✇✐✇✐✳❤✉✲❜❡r❧✐♥✳❞❡ ❤tt♣✿✴✴❝❛s❡✳❤✉✲❜❡r❧✐♥✳❞❡

▼♦t✐✈❛t✐♦♥ ✶✲✶ ❚✐♠❡ ✈❛r✐❛t✐♦♥s ♦❢ ❞♦✇♥s✐❞❡ r✐s❦ ✐♥ ●✼ ❉♦✇♥s✐❞❡ r✐s❦ ✐s ♠❡❛s✉r❡❞ ❜② t❤❡ ✾✾✪ ❱❛❘ ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ ❦❡r♥❡❧ s♠♦♦t❤✐♥❣ ❢♦r t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ❛♥❞ t❤❡♥ ❜♦♦tstr❛♣♣✐♥❣ ❢r♦♠ t❤❡ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t♦r✳ ❚❤❡ ❦❡r♥❡❧ ❞❡♥s✐t② ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ ✐♥ t❤✐s ✜❣✉r❡✳ ❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s

▼♦t✐✈❛t✐♦♥ ✶✲✷ ❘❡s❡❛r❝❤ q✉❡st✐♦♥s ❢♦r ❞♦✇♥s✐❞❡ r✐s❦ ❛♥❞ r❡t✉r♥s ⊡ ❲❤② ❱❛❘❄ ⊡ ❚❤❡ ♦✉t❜r❡❛❦ ♦❢ t❤❡ ✷✵✵✽ ✇♦r❧❞✇✐❞❡ ✜♥❛♥❝✐❛❧ ❝r✐s✐s ⊡ ❲❤❛t ✐s t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❝♦♥t❡♥t ♦❢ ❱❛❘❄ ⊡ ❱❛❘ ✈s✳ ❤✐❣❤❡r ♠♦♠❡♥ts ♦❢ r❡t✉r♥ ♦❢ ❝❞❢ ✭❈♦r♥✐s❤✲❋✐s❤❡r ❡①♣❛♥s✐♦♥✮ ⊡ ▲♦♥❣ ♠❡♠♦r② ♦❢ ✈♦❧❛t✐❧✐t② ❛♥❞ ❱❛❘ ⊡ ❈r♦ss ♠❛r❦❡t ❡✛❡❝ts ♦❢ ✈♦❧❛t✐❧✐t②✴❱❛❘ ⊡ ❚r❛❞❡♦✛ ❤②♣♦t❤❡s✐s s✉♣♣♦rt❡❞❄ ⊡ ▲❡✈❡r❛❣❡ ❡✛❡❝t✴✈♦❧❛t✐❧✐t② ❢❡❡❞❜❛❝❦ s✉♣♣♦rt❡❞❄ ⊡ ▲♦♥❣✲r✉♥ ♦r ❙❤♦rt✲r✉♥ ❡✛❡❝t❄ ⊡ Pr✐❝❡ ❞✐s❝♦✈❡r② ♦❢ ❯❙ ❱❛❘ ✈s✳ ❞♦♠❡st✐❝ ❱❛❘ ❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s

▼♦t✐✈❛t✐♦♥ ✶✲✸ ❱❛❘ ❛s ❛ r❡♣r❡s❡♥t❛t✐✈❡ r✐s❦ ♠❡❛s✉r❡ ❱❛❘ ❛♥❞ ❤✐❣❤❡r ♠♦♠❡♥ts ⊡ ❚❤❡ ❢♦✉rt❤✲♦r❞❡r ❈♦r♥✐s❤✲❋✐s❤❡r ❛♣♣r♦①✐♠❛t✐♦♥ ❛t α % ✲q✉❛♥t✐❧❡ q α α − ✺ z α ) S ✷ α − ✶ ) S α − ✸ z α ) K q α = z α + ( z ✷ ✻ + ( z ✸ ✷✹ − ( ✷ z ✸ ✸✻ ✇❤❡r❡ z α ✐s t❤❡ α ✲q✉❛♥t✐❧❡ ✈❛❧✉❡ ❢r♦♠ t❤❡ st❛♥❞❛r❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ ❛♥❞ S ❛♥❞ K ❛r❡ s❦❡✇♥❡ss ❛♥❞ ❡①❝❡ss ❦✉rt♦s✐s✱ r❡s♣❡❝t✐✈❡❧② ⊡ ❱❛❘ ❛t t❤❡ ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧ ✭✶ ✲ α ✮ V ✶ − α = − σ q α � � � S � K � S ✷ � � � z ✷ ✷ z ✸ ✷ z ✸ = − σ z α + α − ✶ ✻ + α − ✸ z α α − ✺ z α ✷✹ − ✸✻ ❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s

▼♦t✐✈❛t✐♦♥ ✶✲✹ ❊✈✐❞❡♥❝❡ ♦❢ ❧♦♥❣ ♠❡♠♦r② ♦❢ s❡❝♦♥❞ ❛♥❞ ❤✐❣❤❡r ♠♦♠❡♥ts ⊡ ❙t♦❝❦ r❡t✉r♥ ✈❛r✐❛♥❝❡ ❛r❡ ♣❡rs✐st❡♥t ✲ ❉✐♥❣✱ ❊♥❣❧❡ ❛♥❞ ●r❛♥❣❡r ✭✶✾✾✸✮ ❞✐s❝♦✈❡r❡❞ ❛ ❧♦♥❣✲♠❡♠♦r② ♣r♦♣❡rt② ✐♥ st♦❝❦ ♠❛r❦❡t r❡t✉r♥✳ ⊡ ❙❡❝♦♥❞ ♠♦♠❡♥t ♦❢ ❝♦♥❞✐t✐♦♥❛❧ ✈❛r✐❛♥❝❡ s❡r✐❡s ❞✐s♣❧❛②s ❛ ♣❡rs✐st❡♥t ♣❤❡♥♦♠❡♥♦♥ ✲ ❏♦♥❞❡❛✉ ❛♥❞ ❘♦❝❦✐♥❣❡r ✭✷✵✵✸✮✱ ❇❛♥❞✐ ❛♥❞ P❡rr♦♥ ✭✷✵✵✻✮✱ ❛♥❞ ❇♦❧❧❡rs❧❡✈ ❡t ❛❧✳ ✭✷✵✶✸✮ ⊡ ❚❤❡ ❞♦✇♥s✐❞❡ r✐s❦ s❡r✐❡s ❡①❤✐❜✐ts ❛ ❧♦♥❣ ♠❡♠♦r② ✲ ❈❛♣♦r✐♥ ✭✷✵✵✽✮ ❛♥❞ ❑✐♥❛t❡❞❡r ❛♥❞ ❲❛❣♥❡r ✭✷✵✶✹✮ ❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s

▼♦t✐✈❛t✐♦♥ ✶✲✺ ❉♦✇♥s✐❞❡ r✐s❦✲r❡t✉r♥ ✐♥ ❛ ❣❧♦❜❛❧ s❡tt✐♥❣ ⊡ ■s ❯✳❙✳ ♠❛r❦❡t ❛ ♠❛✐♥ ❞r✐✈✐♥❣ ❢♦r❝❡ ❢♦r t❤❡ ●✼ ❞②♥❛♠✐❝ r✐s❦✲r❡t✉r♥ r❡❧❛t✐♦♥❄ ⊡ ❱❛r✐❛♥❝❡ ❛♥❞ ❤✐❣❤❡r ♠♦♠❡♥ts t❡♥❞ t♦ ❝♦✲♠♦✈❡✱ ✇❤❡r❡❛s ✉♥✐✈❛r✐❛t❡ ♠❛r❦❡t ❛♣♣r♦❛❝❤ ✐s ❧✐♠✐t❡❞✳ ⊡ ■♥ t❤❡ ❧♦♥❣ r✉♥✱ ❛ ❝♦✐♥t❡❣r❛t✐♦♥ r❡❧❛t✐♦♥ ❛♣♣❡❛rs ✇✳r✳t ❤✐❣❤❡r ♠♦♠❡♥ts✴❱❛❘ ❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s

❖✉t❧✐♥❡ ❖✉t❧✐♥❡s ✶✳ ▼♦t✐✈❛t✐♦♥ � ✷✳ ❉❛t❛ ❛♥❞ ❡st✐♠❛t❡ ❱❛❧✉❡✲❛t✲❘✐s❦ ✭❱❛❘✮ ✸✳ ❋❈❱❆❘ ♠♦❞❡❧ ✹✳ ❊♠♣✐r✐❝❛❧ r❡s✉❧ts ✭❧♦♥❣✲r✉♥ ✈s✳ s❤♦rt✲r✉♥✮ ✺✳ ■♥❢♦r♠❛t✐♦♥ ❝♦♥t❡♥t ♦❢ ❱❛❘ ✻✳ Pr✐❝❡ ❞✐s❝♦✈❡r② ✭❯❙ ✈s✳ ●✻❄✮ ✼✳ ❈♦♥❝❧✉s✐♦♥ ❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s

❉❛t❛ ❛♥❞ ❱❛❘ ✷✲✶ ✷✳ ❉❛t❛ ❛♥❞ ❡st✐♠❛t✐♥❣ ❞♦✇♥s✐❞❡ r✐s❦ ⊡ ❙t♦❝❦ ♣r✐❝❡ ✐♥❞❡①✿ ❞✐✈✐❞❡♥❞✲❛❞❥✉st❡❞ st♦❝❦ ✐♥❞❡① ✐♥ ❉❛t❛str❡❛♠ ❧❛❜❡❧❡❞ ❛s ❚❖❚▼❑ ⊡ ❈♦✉♥tr✐❡s✿ ❚❤❡ ❯♥✐t❡❞ ❑✐♥❣❞♦♠ ✭❯❑✮✱ ●❡r♠❛♥② ✭●▼✮✱ ❋r❛♥❝❡ ✭❋❘✮✱ ■t❛❧② ✭■❚✮✱ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ✭❯❙✮✱ ❈❛♥❛❞❛ ✭❈❆✮✱ ❛♥❞ ❏❛♣❛♥ ✭❏P✮ ⊡ ❙❛♠♣❧❡ ♣❡r✐♦❞✿ ❙❡♣t❡♠❜❡r ✶✾✾✵ t❤r♦✉❣❤ ❏✉❧② ✷✵✶✸ ⊡ ❋r❡q✉❡♥❝②✿ ❯s❡ ❞❛✐❧② ❞❛t❛ t♦ ❝♦♥str✉❝t ♠♦♥t❤❧② ❱❛❘ ⊡ ❱❛❧✉❡ ❛t ❘✐s❦ ✭❱❛❘✮✿ ♥♦♥♣❛r❛♠❡tr✐❝ ❞❡♥s✐t② ❡st✐♠❛t✐♦♥ ❜② ❦❡r♥❡❧ s♠♦♦t❤✐♥❣ t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ √ ❆♣♣❧② ●❛✉ss✐❛♥ ❦❡r♥❡❧ K h = exp ( − u ✷ / ✷ ) / ✷ π ◮ ❊st✐♠❛t❡ ❛♥ ✐♥t❡❣r❛t❡❞ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t♦r ◮ ❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s

❋❈❱❆❘ ♠♦❞❡❧ ✸✲✶ ✸✳ ❚❤❡ ❢r❛❝t✐♦♥❛❧❧② ❝♦✐♥t❡❣r❛t❡❞ ✈❡❝t♦r ❛✉t♦r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❚❤❡ ❋❈❱❆❘ d ✭♣✮ p � ∆ d z t = α ( µ ′ + β ′ L d z t ) + Γ s L s d ∆ d z t + ε t , t ❂ ✶✱ ✳✳✳✱ ❚ s = ✶ z t ≡ ( V i , t , V j , t , r i , t , r j , t ) ′ ❞❡♥♦t❡ ❛ ✹ × ✶ ✈❡❝t♦r ♣r♦❝❡ss ♣r♦❝❡ss ❝♦♠♣r✐s✐♥❣ t✇♦ ❞♦✇♥s✐❞❡ r✐s❦s✱ V i , t ❛♥❞ V j , t ❛♥❞ t✇♦ st♦❝❦ r❡t✉r♥ ✈❛r✐❛❜❧❡s r i , t ❛♥❞ r j , t ✱ ❛ss✉♠✐♥❣ d = b ▲♦♥❣ r✉♥ ❝♦♠♣♦♥❡♥t � �� �       ˜ ∆ d V i , t βα ✶✶ α ✶✶ α ✶✷ α ✶✸ L d V i , t ∆ d V j , t ˜ βα ✷✶ α ✷✶ α ✷✷ α ✷✸ L d V j , t   = αµ ′ +            ∆ d r i , t ˜ βα ✸✶ α ✸✶ α ✸✷ α ✸✸ L d r i , t      ∆ d r j , t ˜ L d r j , t βα ✹✶ α ✹✶ α ✹✷ α ✹✸ ❙❤♦rt r✉♥ ❝♦♠♣♦♥❡♥t � �� �       L d ∆ d V i , t Γ ✶ , ✶✶ Γ ✶ , ✶✷ Γ ✶ , ✶✸ Γ ✶ , ✶✹ ε ✶ , t L d ∆ d V j , t Γ ✶ , ✷✶ Γ ✶ , ✷✷ Γ ✶ , ✷✸ Γ ✶ , ✷✹ ε ✷ , t       +  +       L d ∆ d r i , t Γ ✶ , ✸✶ Γ ✶ , ✸✷ Γ ✶ , ✸✸ Γ ✶ , ✸✹ ε ✸ , t      L d ∆ d r j , t Γ ✶ , ✹✶ Γ ✶ , ✹✷ Γ ✶ , ✹✸ Γ ✶ , ✹✹ ε ✹ , t ❉♦✇♥s✐❞❡ r✐s❦ ❛♥❞ st♦❝❦ r❡t✉r♥s