Stochastic Ray Propagation in Stratified Random Lattices – Comparative Assessment of Two Mathematical Approaches
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(89) B2G]F^HKJgLRCELUsG*&efHMFICEZh bMZ`_Eh1JMJMLU FICvm BG]FIbKbgFvU¢PVB qàh HMZ dw~ F^JMLNdwC c\dnmEh~tPdwH}Jg_EhzF[wh qEHMdIqfFIrF^JMLNdwCÿLRC CEdwCnVYECELotPdwHMc4HgFICvmEdwc c\hmELNFvUed 7 A<£ Z1f«V¢1¡" I«T§¿ I¸x-¿G[¢1¡g¢1´À`¢}¿¤a-¸N¢`Àb[ «T¡g¿¤\è©I´¢1«i§À`¥'§¨lVI´À`¢Mlz±r± ¸²§ÀMI«T§¿v¥¥¦ZF)aOl¢l§¨b¡rU-|<dwHMLRCEdEUuJ`F^~RwUèrèaIwUèÈ`è hqJghc2¯àh HU Iww. o[<+B2=G]FIHKJMLNCELTU&G*zeXHMFICEZ hbMZ`_vh JMJMLUsFICEm BºG]F^bMbgFU}PfsF[÷qEHMdIqfFIrF^JMLNdwC LNC F*CEdwCYECELRtPdIHMc. HgFICvmEdwc ~NF^JMJMLNZh^Ued^©f¿ªn¡1 I¸¿«ª@± «i§ÀMI¸¬~ ¿èÀ1§¢1«}¿4lD¤¼¢1¡1§ÀMl+Utu{y®¯ÈUXwI aIIvwUvIww è.
(90) ÷1ý'@# ./ + 1 Èó î=¶u· R Ih1JgZ`_dIt´HMF[{qEHKdwqfFIrFJgLNdIC{LNC\FtPdwYEHlV~ F[IhHMbHgFICEmvdwc ~ FJMJgLRZh > ~Nh1t¨JK_fFICEmæbKLNmEh? FICvm÷JM_Eh'dw¯EbaJ`FIZ ~NhbMmEhCvbMLRJV mELNbaJgHMLR¯EYvJMLNdwC HKh~NF^JgLowh¼JMdÿJM_Eh'HgF^CEmEdwc ~NF^JMJMLNZ h > HKLNw_rJaV_fFICvm bKLNmEh ?1. °²±}³¦´µ. Èó î t · G]F^HKJgLRCErFI~Rh2F^qEqEHMdwFIZ`_ysG]FIbH IdèyZ`_fF^LNCþcF^Jg_Eh c¼FJgLNZFI~N~o}c\dmvh~NLRCEJM_EhæHgF[ qEHKdwqfFIwF^JgLRdwC}JgdzFIHMmEb~Nh1wh ~±N LRC F\baJgHgFJgLRùEhm HgFICvmEdwc ~NF^JMJMLNZh^. °²±}³¦´µ. Èó î { · G]FIH ^dè FIqEqEHKdrFIZ`_ 'G$FIH^dÈ Z`_EFILNC cF^Jg_Eh c¼FJgLNZFI~N~oñcdnmEh ~NLNCv Jg_Eh9HgF[ qEHKdwqfFIwF^JgLRdwC'LNCFæwhCvhHMLRZ+CEdwCYECELRtPdIHMc _fFI~Rt qE~ F^CEhHgF^CEmEdwc ~ F^JKJgLNZ hI. °²±}³¦´µ. °²±}³¦´µ. óÈî. , U X. WY[Z > ?I E. ·. |º_EHKhh1~ F[wh HMb qEHKdIùf~RhIU N ^ IhHKbMYEXb N³&_Eh C J %. J % < U ^, Q % 8 FICEm [, j % FÞ baJgLNcF^JMhm$^FI~RYEhbDdIt < tPdwHsJM_Eh2qvHMdIùf~Rhb > R ? F RF U >ML ? RF U > À? U´F^CEm. óÈî s · eXdIYEHa~ F[wh HMb qEHKdIùf~Nh^U J % U ,Q % 8 U ,[j % FÞ FICEm , Æ% } bKJMLNcF^Jgh m ^FI~RYEhbdIt , U .XWY[Z N ^ wh HMbKYEbIN1&_EhC J % < tPdwHJg_EhqEHMd^ùf~Nh b > R? FRF U >ML ? RFRF U > À? EUfFICvm > ?9 . °²±}³¦´µ. Èó î ¸m· |º_EHMh h1~ F[wh HMb+qEHMd^ùf~Nh^U J % < U ,^Q % mELAª´h HMh CJDIF^~NYEh bsdIt : U[B % F +R+P U ¯ hLNCv > $? Q FICvm > ? Q %G. ý . °²±}³¦´µ. Èó î · |º_EHMh h1~ F[wh HMb+qEHMd^ùf~Nh^U ^, Q JM_Eh+LNCEZ LNmEh CEZh
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(93) L4. L3. L2. L1. q(j) q2 q4 q1 q3. j=r3. j=r2. θ. j=0 j=1 j=2. j=r1. j. ±}³¦´ ·º¶. ro. Eó. aJgdnZ`_fFIbKJMLNZ HMFè}qEHMdwqEFIrF^JMLNdwCþoRhd. ·O»½¼ #¾ ³¿)³}À:ÁÂuÃĦŠd. È.
(94) ±}³¦´ · tro. ·O»½¼ #¾ ³¿)³}À:ÁÂuÃĦŠd. Eó. aJgdnZ`_fFIbKJMLNZ HMFè}qEHMdwqEFIrF^JMLNdwCþoRhd è. +. −. empty half−plane. 0. q1. 0. 1. 1. p. 1. −. P1. LAYER L. Q1. + 1. P1. 1. Q1 r1. 1. −. r +. 1. 2. p. q q. 2. p. 1. − r1+1. + r1+1. P2 LAYER L. Q2. P2. 2. 2. Q − r2. + r2. 2. 3. p. q q. 3. p. − r 2+1. 2. + r 2+1. P3 LAYER L. Q3. P3. 3. + rK−1. LAYER L. PK−1 K−1. − rK−1. QK−1. PK−1. K. + r +1 K−1. − r +1 K−1. q K−1. K−1. p. q. K. p. LAYER L. QK. PK. K. k. +.
(95) ±}³¦´ · {ro. ·O»½¼ #¾ ³¿)³}À:ÁÂuÃĦŠd. Eó. aJgdnZ`_fFIbKJMLNZ HMFè}qEHMdwqEFIrF^JMLNdwCþoRhd. [. 1. −. empty half−plane level 1. 1. −. 0. q2. 1. +. q1. 0. +. 1. p. 1. p. 2. p. 2. −. level 2. 1. q. 2. +. 3. q. 2. p. 3. p. 3. −. level 3. 2. q. 3. +. 4. q. k−2. p. − k−1. + k−1 qk. level k−1. k−2. q. k−1. p. k. p. +. level k. k.
(96) > R? Pr{0->k}. 1. MKV approach MTG approach Experiments. 0.5 0. 0. 4. 8. 12. 16 k. Pr{0->k}. 1. 0. 4. 8. 12. 16 k. Pr{0->k}. 1. 0. 4. 8. 12. 16 k. Pr{0->k}. 1. 32. >ML ?. 20. 24. 28. 32. > À?. 20. 24. 28. 32. > ? MKV approach MTG approach Experiments. 0.5 0. 28. MKV approach MTG approach Experiments. 0.5 0. 24. MKV approach MTG approach Experiments. 0.5 0. 20. 0. 4. ±}³¦´ · Æro. 8. Eó. 12. 16 k. 20. 24. 28. 32. aJgdnZ`_fFIbKJMLNZ HMFè}qEHMdwqEFIrF^JMLNdwCþoRhd. ·O»½¼ #¾ ³ ¿)³}À:ÁÂuÃĦŠd. È.
(97) > R? Pr{0->k}. 1. MKV approach MTG approach Experiments. 0.5 0. 0. 4. 8. 12. 16 k. Pr{0->k}. 1. 0. 4. 8. 12. 16 k. Pr{0->k}. 1. 0. 4. 8. 12. 16 k. Pr{0->k}. 1. 32. >ML ?. 20. 24. 28. 32. > À?. 20. 24. 28. 32. > ? MKV approach MTG approach Experiments. 0.5 0. 28. MKV approach MTG approach Experiments. 0.5 0. 24. MKV approach MTG approach Experiments. 0.5 0. 20. 0. 4. ±}³¦´ · sro. 8. Eó. 12. 16 k. 20. 24. 28. 32. aJgdnZ`_fFIbKJMLNZ HMFè}qEHMdwqEFIrF^JMLNdwCþoRhd. ·O»½¼ #¾ ³ ¿)³}À:ÁÂuÃĦŠd. è.
(98) r4o. <δ>. <δ>. ·/»Í¼ -¾ ³¿!³xÀ,ÁÂ\à ĦŠd. vó. È. lnJgdnZ`_fFIbaJgLRZ HgF['qEHKdwqfFIrFJgLNdICþRoed. 312. 121. 313. 123. 314. 124. 321. 131. 132 Profile. É. Ç¦È 323 Profile. 324. 134. 142. 341. 342. MKV approach MTG approach. 141. MKV approach MTG approach. 343. 143. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 412. 212. 413. 213. 414. 214. 421. 231. 232 Profile. 423 Profile. ÇÊ. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. <δ> <δ>. ±}³´ ·^¸ É. É. É. ÇÌ. ǦË. 424. 234. 242. 431. 432. MKV approach MTG approach. 241. MKV approach MTG approach. 434. 243.
(99) 6. MKV approach MTG approach. 5. ∆. 4 3 2 1 0. 15. 30. 45. 60. θ. ±}³´ · r4o. vó. lnJgdnZ`_fFIbaJgLRZ HgF['qEHKdwqfFIrFJgLNdICþRoed. ·/»Í¼ -¾ ³ ¿!³xÀ,ÁÂ\à ĦŠd. è. 75.
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