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UBC Theses and Dissertations

Primary estimation with sparsity-promoting bi-convex optimization Lin, Tim Tai-Yi 2015

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M2`@Qzb2i ;Tb 7Q` KmHiBTH2 `2KQpHU*?Ti2` 9VXeX S`2b2Mi2/  bi`i2;v iQ bB;MB}+MiHv /2+`2b2 i?2 +QKTmiiBQM iBK2`2[mB`2/ iQ bQHp2 i?2 _Q#mbi 1SaA T`Q#H2K BMbTB`2/ #v 2tBbiBM; KmHiB@H2p2H UM/ KmHiB;`B/V K2i?Q/bX h?Bb Bb K/2 TQbbB#H2 #v 2tTHQBiBM; i?2#HBM/ /2+QMpQHmiBQM bT2+i Q7 _Q#mbi 1SaA- b r2HH b i?2 ?B;?@Q`/2`MmK2`B+H +QKTH2tBiv Q7 i?2 2pHmiBQM M/ ;`/B2Mi +QKTmiiBQM bi2Td*?Ti2` RX AMi`Q/m+iBQMQ7 1SaA b  7mM+iBQM Q7 i?2 bTiBH bKTHBM; Q7 i?2 rp2}2H/ U*?Ti2`8VX3Chapter2Principlesofsurfacemultiplepredictionbyconvolutionh?Bb +?Ti2` 2bi#HBb?2b i?2 i?2Q`2iB+H #+F;`QmM/ M/ +QMi2ti M2+2bb`v7Q` i?2 i?2bBbX h?2 KBM #Q/v Q7 i?Bb rQ`F #mBH/b mTQM rp2}2H/ T`QT;iBQMK2i?Q/b 7Q` KmHiBTH2 T`2/B+iBQM- r?B+? `2 #b2/ QM r2HH@FMQrM T?vbB+H#2?pBQm` Q7 rp2b- #mi BM T`+iB+2 7+BHBii2/ #v b2BbKB+ bm`p2v /2bB;Mbi?i `2 QMHv #2+QKBM; 72bB#H2 M/ rB/2bT`2/ BM i?2 Tbi 72r /2+/2bX #`B27 Qp2`pB2r Q7 ?Qr bm#bm`7+2 bi`m+im`2 BM7Q`KiBQM `2 `2Hv2/ #v +iBp2@bQm`+2 b2BbKB+ `2bTQMb2b Bb BM+Hm/2/ 7Q` i?2 #2M2}i Q7 `2/2`b rBi?Qmi Mv#+F;`QmM/ BM b2BbKB+ bm`p2vBM;XA rBHH ;Q BMiQ  KQ`2 /2iBH2/ HBi2`im`2 bm`p2v QM i?2 2`Hv /2p2HQTK2MibQ7 KmHiBTH2 `2KQpH i2+?MB[m2b HHm/2/ iQ BM i?2 T`2pBQmb +?Ti2`- 7QHHQr2/#v  /2b+`BTiBQM Q7 KQ`2 +QMi2KTQ``v KQ/2HHBM;@#b2/ K2i?Q/b i?i `22M#H2/ #v `2+2Mi +[mBbBiBQM /2bB;Mb- 2tT`2bbBM; i?2K mbBM; Ki?2KiB+HBMi2;`H `2HiBQMbX h?Bb /Bb+mbbBQM +mHKBMi2b BM i?2 rp2 2[miBQM@#b2/KmHiBTH2 KQ/2HHBM; i2+?MB[m2b r?B+? KF2 mT i?2 T?vbB+H #bBb Q7 i?2 1SaAK2i?Q/- M/ iQm+?2b QM bQK2 Q7 Bib BMi2`2biBM; `2HiBQMb?BTb rBi? i?2 #HBM//2+QMpQHmiBQM T`Q#H2KX 6BMHHv- r2 HQQF i i?2 b?Q`i+QKBM;b Q7 i?2 +m`@`2Mi 1SaA K2i?Q/b- 7`QK r?B+? i?2 bm#b2[m2Mi +?Ti2`b BM i?Bb i?2bBb `2KQiBpi2/ M/ QmiHBM2/XN*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM2.1 Exploration seismologyh?2 b2BbKB+ `2~2+iBpBiv iBK2@b2`B2b Bb i?2 i?2Q`2iB+H `2bTQMb2 Q7 i?2 bm#bm`@7+2 iQ  T2`72+iHv BKTmHbBp2 b2BbKB+ bQm`+2 Q7 FMQrM KTHBim/2X 6B;m`2 kXR/2TB+ib i?2 `2HiBQMb?BT #2ir22M bm#bm`7+2 bi`m+im`2- `2~2+iBpBiv b2`B2b- M/b2BbKB+ /iXq?2M b2BbKB+ rp2b K22i M BMi2`7+2 #2ir22M bm#bm`7+2 Ki2`BHb rBi?/Bz2`BM; 2HbiB+ T`QT2`iB2b- T`i Q7 i?2 2M2`;v +``B2/ #v i?2 rp2b rBHH #2`2~2+i2/ M/ 2p2MimHHv T`QT;i2/ #+F iQ i?2 bm`7+2X h?2 7`+iBQM Q72M2`;v `2~2+i2/ i 2+? BMi2`7+2 /2T2M/b QM i?2 `2~2+iBQM +Q2{+B2Mi UQ`dz`2~2+iBpBivǴV Q7 i?2 BMi2`7+2- r?B+? BM im`M Bb /2i2`KBM2/ #v i?2 /Bz2`2M+2 BM+QmbiB+ BKT2/M+2 Q7 i?2 irQ Ki2`BHbX "Qi? i?2 bi`2M;i? Q7 i?Bb `2~2+i2/bB;MH M/ Bib ``BpH iBK2 +M #2 mb2/ iQ 7m`i?2` BM/m+2 BM7Q`KiBQM Q7 i?2bm#bm`7+2 ;2QHQ;vXhvTB+HHv i?2 bm#bm`7+2 BMi2`7+2b `2 bbmK2/ iQ #2 Q7 x2`Q@Q`/2`- r?B+?`2bmHib BM M BKTmHbBp2 `2~2+iBQM 7Q` M BKTmHbBp2 BM+B/2Mi 2p2MiX h?2 ``BpHiBKBM; Q7 i?2b2 `2~2+iBQMb iQ i?2 bm`7+2 Bb  i?mb 7mM+iBQM Q7 #Qi? i?2bm#bm`7+2 rp2 p2HQ+Biv M/ i?2 HQ+iBQM Q7 i?2 BMi2`7+2bX h?Bb `2HiBQMb?BTBb 2tTHQBi2/ BM bm#b2[m2Mi BK;BM; bi2Tb iQ T`Q/m+2  bi`m+im`H /2TB+iBQMQ7 i?2 bm#bm`7+2Xa2BbKB+ /i T`Q+2bbBM; Bb i?2 T`+iB+2 Q7 T`2T`BM; T?vbB+HHv +[mB`2/b2BbKB+ /i BM i?2 }2H/ 7Q` i?Bb ivT2 Q7 b2MbBiBp2 BM72`2M+2 ibFX P#iBMBM; MQBb2@7`22- KTHBim/2@+Q``2+i b2BbKB+ `2~2+iBpBiv iBK2@b2`B2b Bb `2;`/2/b i?2 ;QH Q7 b2BbKB+ T`Q+2bbBM;X >BM/2`BM; i?Bb ;QH `2 KMv T`+iB+H7+iQ`b- #mi i?2 QM2b KQbi T2`iBM2Mi iQ i?Bb i?2bBb BM+Hm/2 i?2 T`2b2M+2 Q7#Qi? bm`7+2@`2Hi2/ M/ BMi2`MH KmHiBTH2b- b r2HH b #M/@HBKBi2/ bB;Mim`2Q7 i?2 `iB}+BH b2BbKB+ bQm`+2bX h?2b2 `2 #Qi? 7+iQ`b i?i Q#7mb+i2 i?2B/2H `2~2+iBpBiv b2`B2b mM/2`HvBM;  b2BbKB+ `2+Q`/BM; M/ KF2b ++m`i2b2BbKB+ bm`p2vBM; M BM?2`2MiHv /B{+mHi ibF- 2p2M BM i?2 #b2M+2 Q7 QmibB/2MQBb2X.m2 iQ T?vbB+H HBKBiiBQMb BM 2M2`;v QmiTmi- `iB}+BH b2BbKB+ bQm`+2bmb2/ BM 2tTHQ`iBQM +MMQi #2 T2`72+iHv BKTmHbBp2X AMbi2/ i?2v ivTB+HHv?p2  M``Qr 7`2[m2M+v #M/rB/i? /2i2`KBM2/ #v i?2 ivT2 Q7 2[mBTK2Mimb2/- ivTB+HHv #QmM/2/ #Qp2 i 3y >x M/ #2HQr #v Ry >x- iFBM; QM rp2H2i@HBF2 TT2`M+2 BM iBK2X h?2 `2+Q`/2/ `2~2+iBpBiv b2`B2b 7`QK bm+? bQm`+2 rBHH i?mb #2 /BbiQ`i2/ #v i?2 rB/i? Q7 i?2 bB;Mim`2 rp2H2i M/ BibKTHBim/2X h?Bb HQr2`b i?2 `2bQHpBM; TQr2` 7Q` i?BM bi`m+im`2b- M/ Q#b+m`2bi?2 KBMmi2 /Bz2`2M+2b #2ir22M `2~2+i2/ M/ `27`+i2/ 2p2MibX AM 6B;m`2 kXR-Ry*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM0100200300400500600700-6-4-202460100200300400500600700-6-4-202460100200300400500600700-6-4-202460100200300400500600700-6-4-202460100200300400500600700-6-4-202460100200300400500600700-6-4-202460 100 200 300 400 500 600 700-10-50510geologic sectioninterfaces(–) (+)0 100 200 300 400 500 600 700-10-50510recorded signalGreen's functionreflectivity seriesdepthtime6B;m`2 kXR 1tKTH2 Q7 `2~2+iBpBiv b2`B2b K2+?MBbKX G27i b2+iBQMb b?QrbbBt /BbiBM+i ;2QHQ;B+H Hv2`b UBM+Hm/BM;  ri2` +QHmKMV rBi? }p2 /BbiBM+iBMi2`7+2bX JB//H2 b2+iBQM b?Qrb i?2 `2~2+iBpBiv +Q2{+B2Mi BM/m+2/ #v +QM@i`bi BM +QmbiB+ BKT2/M+2 #2ir22M i?2 Hv2`bX _B;?i b2+iBQM b?Qrb ?Qr bBM;H2@`2+2Bp2` dzi`+2Ǵ Q7 b2BbKQ;`K Bb T`Q/m+2/ mM/2` M 2ti2`MH bQm`+2rBi?  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T`Q/m+2/ pX h?mb- g +M #2 b22M b i?2 T`BK`vrp2}2H/ po +QKTQM2Mi Q7 p i?i Bb HbQ /2+QMpQHp2/ 7`QK i?2 bQm`+2 rp2H2iq- rBi? i?2 `2HiBQM po(x,ω;xb`+) = g(x,ω;xb`+)q(ω)X hQ F22T i?Bb i?2bBbKM;2#H2 BM b+QT2- A bbmK2 i?i HH BM/2T2M/2Mi b2BbKB+ 2tT2`BK2Mib i2+? bQm`+2 HQ+iBQM ?p2 `Qm;?Hv i?2 bK2 bQm`+2 7mM+iBQM BM iBK2X A HbQ`272` iQ g b i?2 :`22MǶb 7mM+iBQM BMbi2/ Q7 Bib MQ`KH /2`BpiBp2- M/ bBKTHvbbmK2 i?i HH bm`7+2 BMi2;`Hb BMpQHpBM; Bi BM+Hm/2  r2HH@FMQrM Q#HB[mBiv7+iQ`XR3*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM⇤=Primary All surface multiplesmultiple order1 2 3 4 ... ∞1 2 3 4 ... ∞6B;m`2 kX8 *QMpQHpBM;  rp2}2H/ +QMiBMBM; bm`7+2 KmHiBTH2b Q7 /Bz2`2MiQ`/2` UmT iQ BM}MBivV rBi? i?2 T`BK`v `vTi? 2bb2MiBHHv T`Q/m+2b dzb?B7ibǴHH Q`/2`b Q7 bm`7+2 KmHiBTH2b mT #v QM2 UBX2X- Rbi Q`/2` KmHiBTH2b #2+QK2bkM/ Q`/2`- 2i+VX S`BK`v rp2}2H/b +M 2bb2MiBHHv #2 pB2r2/ BM i?Bb rvb  dzx2`Qi? Q`/2`Ǵ bm`7+2 KmHiBTH2X *QMpQHmiBQM Q7 i?2 Q#b2`p2/ /irBi? i?2 T`BK`v `vTi? i?2`27Q`2 T`Q/m+2b HH bm`7+2 KmHiBTH2b Q7 Q`/2` RM/ #Qp2- M/ H2p2b MQ T`BK`B2bX h?Bb 2z2+i 7Q`Kb i?2 T`BM+BTH2 #2?BM/KmHiBTH2 T`2/B+iBQM #v rp2}2H/ T`QT;iBQMX2.3.2 Propagationbasedonestimatedsubsurfacemodelsq2 b22 i?i BM Q`/2` iQ +QKTH2i2Hv ;2M2`i2 HH i?2 bm`7+2 KmHiBTH2b +Q``2+iHv-r2 M22/ iQ FMQr i?2 `vTi? Q7 i?2 T`BK`v rp2}2H/X >Qr2p2`- i?Bb FBM/ Q7/2iBH2/ bm#bm`7+2 BM7Q`KiBQM Bb 2t+iHv r?i r2 ?p2 ?QT2/ iQ T`Q#2 rBi?i?2 b2BbKB+ bm`p2v BM i?2 }`bi TH+25 A7 QM2 QMHv rBb?2/ iQ `2KQp2 i?2 b?HHQr@Hv2` `2p2`#2`iBQMb Ubm+? b BM i?2 ri2` +QHmKMV-  bmBi#H2 TT`QtBKiBQMQ7 i?2 b?HHQr Hv2` +M BM bQK2 +b2b #2 2biBKi2/ r2HH 2MQm;? iQ BKTHB+BiHv;2M2`i2  bmBi#H2 T`BK`v T`QT;iBQM F2`M2H g UivTB+HHv #v bQHpBM; i?2rp2 2[miBQM 7Q` i?2 2biBKi2/ K2/BmKVX h?Bb K2i?Q/ Bb ivTB+HHv FMQrMb dzKmHiBTH2 T`2/B+iBQM #v rp2}2H/ 2ti`TQHiBQMǴ U"2`Mi? M/ aQMM2HM/-RN3jc "2``v?BHH M/ EBK- RN3ec G2pBM- RN3dc qB;;BMb- RN33c GQFb?iMQp-kyyyVXRN*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQMM2+/QiHHv- i?2 rp2}2H/ 2ti`TQHiBQM K2i?Q/ +M #2 p2`v /B{+mHi iQiiBM  ?B;? /2;`22 Q7 ++m`+v 7Q` KMv bm`p2v }2H/b rBi? +QKTH2t M2`@bm`7+2 ;2QK2i`vc Q7i2M Bi Bb T`Q#H2KiB+HHv b2MbBiBp2 iQ 2p2M bHB;?i 2``Q`bBM i?2 bm#bm`7+2 KQ/2HX 6m`i?2`KQ`2- Dmbi KQ/2HHBM; i?2 b?HHQr `vTi?brQmH/ MQi #2 biBb7+iQ`v 7Q` T`QD2+ib i?i M22/ iQ b22 p2`v /22T BMiQ i?21`i?X AM +QMi2KTQ``v mb;2- mMH2bb BM p2`v b?HHQr K`BM2 +[mBbBiBQMb-i?2 rp2}2H/ 2ti`TQHiBQM K2i?Q/ Bb H`;2Hv bmTTHMi2/ BM TQTmH`Biv #vr?i `2 FMQrM b dz/i@/`Bp2M K2i?Q/bǴ /2b+`B#2/ #2HQr- r?B+? `2 Km+?KQ`2 bi`B;?i7Q`r`/ r?2M #2BM; TTHB2/ iQ `#Bi``v bm`p2vbX2.3.3 PropagationbasedonrecordeddataAMbi2/ Q7 2tTHB+BiHv T2`7Q`KBM; rp2}2H/ 2ti`TQHiBQM #v +QMbi`m+iBM; KQ/2H M/ bQHpBM; i?2 rp2 2[miBQM- r2 +M mb2 i?2 /i Bib2H7 b M T@T`QtBKiBQM Q7 i?2 :`22MǶb 7mM+iBQM Q7 i?2 rp2 2[miBQMX h?Bb Bb pHB/ B7 i?2`2~2+iBM; bm`7+2 Bb +Q@HQ+i2/ rBi? i?2 /2i2+iQ` bm`7+2- r?B+? Bb ;2M2`HHvi`m2 7Q` bm`7+2 KmHiBTH2bX h?2 ;2M2`H +Hbb Q7 K2i?Q/b i?i mb2b i?2 /iBib2H7 b i?2 KmHiBTH2@;2M2`iBQM rp2}2H/ T`QT;iQ` Bb ;2M2`HHv FMQrM bdz/i@/`Bp2M K2i?Q/bǴXh?Bb b+?2K2 +M }`bi #2 7QmM/ BM Mbi2v M/ L2rKM URNeeV- r?B+?MQi2/ i?i i?2 miQ+QMpQHmiBQM Q7  b2BbKB+ i`+2b +QMiBMBM; QMHv T`BK`B2bi`Mb7Q`Kb Bi BMiQ  i`+2 +QMiBMBM; QMHv }`bi Q`/2` bm`7+2 KmHiBTH2bX +QKTH2i2 KQ/2H 7Q` bm`7+2 KmHiBTH2b #b2/ QM i?Bb T`BM+BTH2 Bb /2b+`B#2/ #vE2MM2ii URNdNV 7Q` R. 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BbivTB+HHv Kmi2/ Q` /Bb+`/2/ bBM+2 QMHv bm#bm`7+2 `2~2+iBQMb M/ `27`+@iBQMb `2 Q7 BMi2`2bi 7Q` 2tTHQ`iBQM T`QD2+ib UwBQHFQrbFB 2i HX- RNN3- RNNNVXh?Bb 2M#H2b  /2+QMpQHmiBQM iQ #2 /2i2`KBMBbiB+HHv #mBHi BMiQ i?2 /TiBp2bm#i`+iBQM M/ +M BKT`Qp2 i?2 `2HB#BHBiv Q7 i?2 bQHmiBQM bQK2r?iXkR*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQMPi?2`b /pQ+i2/ M BKT`Qp2/ b2H2+iBQM Q7 T`BQ`b 7Q` i?2 `2bB/m2 rp2@}2H/ U`2+Q`/2/ /i KBMmb i?2 KmHiBTH2 KQ/2HV rBi? r?i +M #2 pB2r2/ b"v2bBM@#b2/ TT`Q+?2b iQ /TiBp2 bm#i`+iBQMX PM2 Q7 i?2 2`HB2bi 2t@KTH2 Bb aTBix URNNNV r?Q BMi`Q/m+2/  b+?2K2 #b2/ QM Tii2`M `2+Q;MBiBQMmbBM; bTiBH T`2/B+i#BHBiv Q7 i?2 T`BK`v M/ KmHiBTH2 `2~2+iBQMbXMQi?2` bm++2bb7mH +Hbb Q7 K2i?Q/b mb2b i?2 bT`bBiv BMbi2/ Q7 i?2 KBM@BKmK 2M2`;v +`Bi2`B 7Q` i?2 T`BK`v rp2}2H/ U:mBiiQM M/ o2`b+?mm`-kyy9V- mbmHHv mM/2` bQK2 bT`bB7vBM; rp2}2H/ `2T`2b2MiiBQMb bm+? b *m`@p2H2ib U>2``KMM 2i HX- kyydc qM; 2i HX- kyy3c >2``KMM 2i HX- kyy3cL22HKMB 2i HX- kyy3VX _2Hi2/Hv- KQ`2 bQT?BbiB+i2/ K2i?Q/b r2`2 HbQ`2+2MiHv /2p2HQT2/ mbBM; i?2 BM/2T2M/2Mi +QKTQM2Mi MHvbBb UA*V #v E@THM M/ AMMM2M Ukyy3VX a2p2`H mi?Q`b ?p2 #mBHi mTQM i?Bb 7`K2rQ`FUGm M/ GBm- kyyNc .QMMQ- kyRRV BM+Hm/BM;  +b2 i?i 2tTHQBib i?2 bi`m+im`2Q7 i?2 rp2 2[miBQM BM j. #v bm#i`+iBM; 2MiB`2 b2BbKB+ pQHmK2b i QM+2UGB M/ Gm- kyRjVX2.4.2 3D SRMEAi b?QmH/ }`bi #2 MQi2/ i?i i?2 Ki?2KiB+H 7Q`KHBbK #Qp2 TTHB2b 7mHHviQ  j. +[mBbBiBQM ;2QK2i`vX >Qr2p2`- Bi T`Qp2/ p2`v +?HH2M;BM; iQ bKTH2 bm`7+2 i  }M2 2MQm;? ;`B/ iQ biBb7v HBbBM;@7`22 `2[mB`2K2Mib 7Q` i?2b2rp2}2H/ T`QT;iBQM K2i?Q/b 7Q` KmHiBTH2 KQ/2HHBM;X i H2bi QM2 bTiBH/B`2+iBQM rBHH #2 Km+? KQ`2 TQQ`Hv bKTH2/- /m2 iQ i?2 Qp2`r?2HKBM; K@DQ`Biv Q7 b2BbKB+ /2i2+iQ`b U;2QT?QM2b M/ ?v/`QT?QM2bV #2BM; 2K#2//2/BMiQ +#H2bX .2bTBi2 i?2 bKTHBM; +?HH2M;2b BMpQHp2/- KMv ii2KTib ?p2#22M K/2 iQ KF2 a_J1 T`+iB+H 7Q` j. /i- iQ //`2bb 7mM/K2MiHBM++m`+B2b Q7 i?2 k. K2i?Q/ UJibQM M/ *Q``B;M- kyyyVX JMv rvb `2bm;;2bi2/ iQ Qp2`+QK2 i?2 bKTHBM; /2}+B2M+B2b i?i Bb ivTB+H Q7 j. bm`p2vbBM i?2 +`Qbb@HBM2 /B`2+iBQMX UL2Fmi- RNN3c "B2`bi2F2`- kyyRc pM .2/2K M/o2`b+?mm`- kyy8c .QMMQ 2i HX- kyRyVX  ;QQ/ bm`p2v Q7 i?2 +m``2Mi bBimiBQMBM j. Bb .`;Qb2i 2i HX UkyRyVXb  `2bmHi Q7 i?2 #Qp2 /2p2HQTK2Mib- M/ /m2 iQ i?2 ;2M2`H 2b2 Q7TTHvBM; i?2 K2i?Q/- a_J1 iQ/v 2MDQvb rB/2bT`2/ +QKK2`+BH bm++2bb-/2bTBi2 Bib BM?2`2Mi HBKBiiBQM Q7 `2HvBM; QM bQT?BbiB+i2/ bm#i`+iBQM bi2TbXb Q7 MQi HQM; ;Q 7`QK i?2 iBK2 Q7 r`BiBM;- Bi Bb biBHH `2+Q;MBx2/ BM i?2 b2BbKB+bm`p2vBM; BM/mbi`v b i?2 KQbi ++m`i2 bm`7+2 KmHiBTH2 `2KQpH K2i?Q/7Q` KQbi bm`p2v T`QD2+ibX Uo2`b+?mm`- RNNkc o2`b+?mm` M/ "2`F?Qmi- RNNdc.`;Qb2i M/ C2`BÍ2pBÉ- RNN3c o2`b+?mm` M/ S`2BM- RNNNc >/B/B 2i HX- RNNNVXkk*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM2.5 Estimation of primaries by sparseinversionAM :mBiiQM M/ o2`b+?mm` Ukyy9V M/ pM :`Q2M2biBDM M/ o2`b+?mm` Ukyy3V-Bi rb b?QrM i?i 7Q` /TiBp2 bm#i`+iBQM i?2 +Q``2+i `2bmHi Bb Q7i2M vB2H/2/#v i?2 KBMBKBx2` Q7  bT`bBiv K2bm`2 QM i?2 T`BK`v rp2}2H/X h?Bb Q#b2`@piBQM H2/ iQ i?2 `2HBxiBQM i?i T2`?Tb /B`2+iHv BMp2`iBM; 7Q`  /2+QMpQHp2/p2`bBQM Q7 i?2 T`BK`v rp2}2H/ +QmH/ #2 M Hi2`MiBp2 iQ BKT`Qp2 mTQMbQK2 Q7 i?2 7mM/K2MiH HBKBiiBQMb Q7 a_J1X  +QKTH2i2 K2i?Q/ rb 2p2M@imHHv BMi`Q/m+2/ BM pM :`Q2M2biBDM M/ o2`b+?mm` UkyyNV b  T`Q+2/m`2+HH2/ i?2 1biBKiBQM Q7 S`BK`B2b #v aT`b2 AMp2`bBQM U1SaAVXh?2 Q#D2+iBp2 Q7 1SaA Bb iQ /B`2+iHv bQHp2 7Q`  rp2}2H/ F2`M2H g i?i +MbBKmHiM2QmbHv 2tTHBM #Qi? i?2 KmHiBTH2b M/ i?2 T`BK`B2b, i?2 KmHiBTH2bi?`Qm;? 2tT`2bbBQM kXR- b r2HH b i?2 T`BK`B2b mbBM;  DQBMiHv@2biBKi2/rp2H2iX h?Bb 7mM/K2MiH /Bz2`2M+2 rb b?QrM iQ #2 2MQm;? iQ //`2bb/2}+B2M+B2b BM i?2 +[mBbBiBQM Q7 p #v m;K2MiBM; i?2 7mM+iBQMHb BMpQHp2/BM 1SaAǶb p`BiBQMH T`Q#H2K BM p`BQmb rvbXh?2 B/2 Q7 /B`2+iHv 2biBKiBM; i?2 T`BK`v rp2}2H/ Bb MQi M2rX a2p2`Hmi?Q`b ?p2 T`2pBQmbHv 2tTHQ`2/ rvb iQ 2biBKi2 2Bi?2` i?2 bm`7+2@7`22:`22MǶb 7mM+iBQM Q` i?2 T`BK`v rp2}2H/ /B`2+iHv UpM "Q`b2H2M 2i HX- RNNec"B2`bi2F2`- kyyRc qM;- kyy9VX h?2 KQbi 7KQmb Q7 i?2b2 Bb BMi`Q/m+2/ BMKmM/b2M UkyyRV- r?B+? BMbi2/ Q7 ?pBM; iQ HbQ 2biBKi2 i?2 rp2H2i-BMbi2/ `2HB2b QM mbBM; i?2 `2+Q`/2/ /B`2+i rp2 b T`QTQb2/ #v wBQHFQrbFB2i HX URNNNVX >Qr2p2`- b Q7 i?2 KQbi `2+2Mi ii2KTib `2TQ`i2/ BM JD/ƖbFB2i HX UkyRRV- i?2 /B`2+i rp2 biBHH M22/b iQ #2 `2+QMbi`m+i2/ bvMi?2iB+HHv7`QK K2bm`2K2MibX6QHHQrBM; 2[miBQM kXR- r2 ?p2 M 2tT`2bbBQM i?i KTb g iQ #Qi? i?2T`BK`v M/ i?2 bm`7+2@`2Hi2/ KmHiBTH2 2p2Mib BM i?2 iQiH rp2}2H/ pX A`2T`2b2Mi i?Bb KQ/2H b  iBK2@/QKBM QT2`iQ` M(g, q; p) i?i Bb  7mM+iBQMQ7 g M/ i?2 bQm`+2 rp2H2i q- T`K2i2`Bx2/ #v i?2 /i,M(g, q; p) := F−1ω[g(x,ω;xb`+)q(ω) +∫Sg(x,ω;x′)r(x,x′)p(x′,ω;xb`+) dx′]UkXkV= po(x, t;xb`+) + pm(x, t;xb`+)= p(x, t;xb`+),kj*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQMr?2`2 F−1ω `2T`2b2Mib i?2 +QMiBMmQmb BMp2`b2 6Qm`B2` i`Mb7Q`KX b 6B;@m`2 kXe b?Qrb BM R. KQ/2Hb- FMQrBM; i?2 i`m2 :`22MǶb 7mM+iBQM ;`2iHv BK@T`Qp2b i?2 ++m`+v Q7 i?2 KQ/2HH2/ KmHiBTH2bX2.5.1 DiscretizednotationAM T`+iB+2- 1SaA 7QHHQrb i?2 /Bb+`2iBx2@i?2M@QTiBKBx2 T`/B;K- r?B+? i@i2KTib iQ bQHp2 /B`2+iHv 7Q` /Bb+`2iBx2/ `2T`2b2MiiBQMb g M/ q U/Bb+`2iBx2/2MiBiB2b rBHH #2 /2MQi2/ #v #QH/ 7QMiV mbBM; MmK2`B+ H;Q`Bi?KbX q2 `2 i?mbQMHv +QM+2`M2/ rBi? i?2 /Bb+`2iBx2/ p2`bBQM Q7 2tT`2bbBQM kXkX bbmKBM; `2;mH`Bx2/ bKTHBM; Q7 bQm`+2 M/ `2+2Bp2` +QQ`/BMi2b- i?2 bm`7+2 BMi2@;`H BM 2tT`2bbBQM kXR +M HbQ #2 r`Bii2M b bBKTH2 Ki`Bt KmHiBTHB+iBQMb#2ir22M KQMQ+?`QKiB+ bHB+2b Q7 /i pQHmK2b- 7QHHQrBM; i?2 /2iBH@?B/BM;MQiiBQM Q7 "2`F?Qmi M/ SQ URN3kVX h?`Qm;?Qmi i?Bb i?2bBb A rBHH /2MQi2i?2 KQMQ@7`2[m2M+v Ki`Bt pB2r Q7  rp2}2H/ #v r`BiBM; i?2 bvK#QH BMmTT2`@+b2 H2ii2`b U/`QTTBM; i?2 BKTHB2/ 7`2[m2M+v /2T2M/2M+2 7Q` #`2pBivVX6Q` 2tKTH2-  KQMQ@7`2[m2M+v /i Ki`Bt 7Q` i?2 rp2}2H/ p UQ7 bBx2nt × nrcv × nsrcV Bb r`Bii2M bBKTHv b P UQ7 bBx2 nrcv × nsrcVX A /?2`2 iQ i?2+QMp2MiBQM Q7 `Qr U+QHmKMV BM/B+2b Q7  /i Ki`Bt +Q``2bTQM/BM; iQ i?2/Bb+`2iBx2/ `2+2Bp2` UbQm`+2V TQbBiBQMbXh?2 /Bb+`2iBx2/ 7Q`r`/ KQ/2H 7Q` 1SaA U2[miBQM kXkV Bb i?mbM(g,q;p) = F−1ω[Mω(G,Q;P)], UkXjVMω(G,Q;P) := GQ+GRP.h?2 QT2`iQ` F−1ω Bb i?2 UT//2/V /Bb+`2i2 BMp2`b2 6Qm`B2` i`Mb7Q`K i?iT`2p2Mib +QMpQHmiBQMH r`T`QmM/ 2z2+ib BM i?2 iBK2 /QKBMX h?2 i2`KGQ T`Q/m+2b i?2 T`BK`v 2p2Mib- r?BH2 i?2 GRP i2`K T`Q/m+2b bm`7+2KmHiBTH2bX LQi2 i?i i?2 /Bb+`2iBx2/ MQiiBQM HHQrb 7Q`  KQ`2 `#Bi``vbQm`+2 7mM+iBQM +QKT`2/ iQ i?2 +QMiBMmQmb 2tT`2bbBQM- #mi A `2+QM+BH2 i?2irQ #v T`K2i2`BxBM; Q = q(ω)IX aBKBH`Hv- i?2 `2~2+iBpBiv QT2`iQ` BbbbmK2/ BbQi`QTB+ rBi? `2~2+iBQM +Q2{+B2Mi −1 b #Qp2- #v b2iiBM; R = −IX6Q` +H`Biv- };m`2 kXd b?Qrb  b?Qi@;i?2` Uxb`+ }t2/ i  T`iB+mH` TQBMiV7Q` i?2 p`BQmb rp2}2H/b K2MiBQM2/ M/ T`Q/m+2/ BM i?2 #Qp2 `2HiBQMXaBKmHiM2QmbHv bQHpBM; 7Q` g b r2HH b i?2 bQm`+2 rp2H2i 7Q` rBHH BM;2M2`H /KBi MQM@mMB[m2 bQHmiBQMb- MHQ;Qmb iQ MQM/2i2`KBMBbiB+ /2+QMpQ@HmiBQM T`Q#H2Kb rBi? M mMFMQrM #Hm` F2`M2H UA rBHH /Bb+mbb KQ`2 QM i?BbiQTB+ BM b2+iBQM kX8XjVX h?mb- Q#iBMBM;  mb27mH bQHmiBQM `2[mB`2b //BiBQMHk9*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.500.5Surface-free Greens function0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.500.5Observed pressure data0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.500.5True surface multiples0 0.2 0.4 0.6 0.8 1 1.2 1.4-202SRME prediction (of surface multiples)time (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.500.5EPSI prediction (of data)6B;m`2 kXe aBM;H2 i`+2 UR.V bm`7+2 KmHiBTH2 KQ/2H Q7 M mM/2`HvBM; irQ@`2~2+iQ` 2p2Mi b2BbKB+ bB;MHX *QMpQHpBM; i?2 Q#b2`p2/ T`2bbm`2 /i rBi?Bib2H7 Ua_J1 T`2/B+iBQMV T`Q/m+2b  rp2}2H/ i?i Bb bBKBH` iQ i?2 i`m2bm`7+2 KmHiBTH2 #mi Bb 2``QM2Qmb BM #Qi? i?2 KTHBim/2 M/ i?2 Qp2`HHb?T2 Q7 i?2 KmHiBTH2 2p2MibX h?2 1SaA `2HiBQM BM im`M Bb #H2 iQ 2tTHBMi?2 r?QH2 /i ++m`i2Hv #v miBHBxBM; i?2 :`22MǶb 7mM+iBQMXk8*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM`2;mH`BxiBQM QM i?2 mMFMQrMbX h?2 BMbB;?ib BM pM :`Q2M2biBDM M/ o2`@b+?mm` Ukyy3V bm;;2bib i?i i?Bb `2;mH`BxiBQM b?QmH/ #2 KBMBKmK bT`bBivQM gX h?Bb Bb HbQ DmbiB}2/ #v i?2 7+i i?i  /Bb+`2iBx2/ `2T`2b2MiiBQM Q7 i?2:`22MǶb 7mM+iBQM g UBM j@.V `2b2K#H2b  rp2}2H/ rBi? BKTmHbBp2 Q` dzbT`b2Ǵrp27`QMibX h?2`27Q`2- i?2 1SaA H;Q`Bi?K BMi`Q/m+2/ BM i?Bb i?2bBb T`Q@TQb2b iQ }M/ i?2 bT`b2bi TQbbB#H2 g UBM i?2 iBK2 /QKBMV i?`Qm;? ℓ0@MQ`KKBMBKBxiBQMX aT2+B}+HHv- Bi bQHp2b i?2 7QHHQrBM; QTiBKBxiBQM T`Q#H2K,ming, q∥g∥0 bm#D2+i iQ f(g,q;p) ≤ σ, UkX9Vf(g,q;p) := ∥p−M(g,q;p)∥2,r?2`2 f(g,q;p) Bb i?2 KBb}i 7mM+iBQMH- M/ i?2 KQ/2HHBM; QT2`iQ`M(g,q;p) Bb b /2}M2/ BM 2tT`2bbBQM kXjX AM i?2 Q`B;BMH 1SaA 7Q`KmHiBQMUpM :`Q2M2biBDM M/ o2`b+?mm`- kyyNV- i?Bb KBMBKmK bT`bBiv bbmKTiBQMQM g Bb 2M7Q`+2/ #v M /@?Q+ ?`/ i?`2b?QH/ QT2`iBQM QM ;`/B2Mi mT/i2b7Q`K2/ iQ KBMBKBx2 f(g,q;p)X h?Bb Bb  b?Q`i+QKBM; i?i A rBHH //`2bb BM*?Ti2` jX2.5.2 ImprovementsofEPSI overSRMEPM2 Q7 i?2 KBM KQiBpiBQMb #2?BM/ i?2 T`QTQbH Q7 1SaA Bb i?i #v 2HBKBMi@BM; i?2 M22/ 7Q` /TiBp2 bm#i`+iBQM /m2 iQ ?pBM;  Km+? KQ`2 ++m`i2KmHiBTH2 KQ/2H- 1SaA rBHH T`Q/m+2 Km+? KQ`2 ++m`i2 T`BK`B2b BM `2br?2`2 T`BK`B2b M/ KmHiBTH2b b2p2`2Hv Qp2`HTX h?2`2 `2 b2p2`H `2TQ`ibi?i b?Qr 7B`Hv +QMpBM+BM; 2tKTH2b QM `2H /i i?i i?Bb Bb BM/22/ i?2+b2 UpM :`Q2M2biBDM M/ o2`b+?mm`- kyyN#c ap2Hb 2i HX- kyRyc "`/KM2i HX- kyRyVXMQi?2` BKTQ`iMi /pMi;2 Q7 1SaA Bb i?i Bi `2HB2b QM  7mHH Bi2`@iBp2 BMp2`bBQMfQTiBKBxiBQM T`Q+2bb 7`QK  T?vbB+H KQ/2H i?i +QMbB/2`b i?2r?QH2 Q#b2`p2/ /iX h?Bb HHQrb mb iQ #mBH/ p`BQmb FBM/b Q7 MQBb2 BMiQ i?27Q`r`/ KQ/2H- M/ +M BM?2`2MiHv `2D2+i //BiBp2 MQBb2 B7 i?2 T`+iBiBQM2`+M pQB/ Qp2`}iiBM;X  `2Hi2/ 2ti2MbBQM Bb i?2 #BHBiv iQ KBiB;i2 KBbb@BM; /i BM i?2 rp2}2H/ r?B+? 7mM/K2MiHHv H2/b iQ 2``QM2Qmb KmHiBTH2T`2/B+iBQM- bBM+2 i?2 BMi2;`iBQM bm`7+2 rQmH/ 2z2+iBp2Hv #2 BM+QKTH2i2HvbKTH2/X h?Bb bT2+i rb HbQ bm++2bb7mHHv /2KQMbi`i2/ BM i?2 #Qp2 `2@TQ`ib- Hi?Qm;? A rBHH T`QTQb2 M Hi2`MiBp2 TT`Q+? BM *?Ti2` 9Xke*?Ti2`kXS`BM+BTH2bQ7bm`7+2KmHiBTH2T`2/B+iBQM#v+QMpQHmiBQMOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4UVOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4U#VOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4U+VOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4U/V6B;m`2 kXd a?Qi ;i?2`b Q7 i?2 p`BQmb rp2}2H/b BMpQHp2/ BM i?Bb TT2`X UV i?2 +QKTH2i2 T`2bbm`2 /i p- U#V i?2i`m2 /Bb+`2i2 bm`7+2@7`22 :`22MǶb 7mM+iBQM gX S`Q/m+BM; i?Bb Bb i?2 T`BK`v Q#D2+iBp2 Q7 i?2 1SaA T`Q#H2KX U+V i?2a_J1 KmHiBTH2 KQ/2H- T`Q/m+2/ #v i?2 2[mBpH2Mi Q7  PRP i2`K- mbBM; i?2 Q#b2`p2/ /i b M 2biBKiBM; Q7 gi?2 T`BK`v `vTi? T`QT;iQ`X h?2 KTHBim/2b M/ ;2M2`H b?T2 Q7 i?2 KmHiBTH2b `2 2``QM2Qmb r?2M +QKT`2/iQ i?2 KmHiBTH2 2p2Mib BM i?2 /iX U/V i?2 1SaA KmHiBTH2 KQ/2H T`Q/m+2/ #v i?2 GRP i2`K BM 2tT`2bbBQM kXj-r?B+? Bb  Km+? KQ`2 ++m`i2 KQ/2H Q7 i?2 bm`7+2 KmHiBTH2- M/ +M #2 /B`2+iHv bm#i`+i2/ 7`QK i?2 /i rBi?biBb7+iQ`v `2bmHibXkd*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM6BMHHv- 1SaA /Q2b MQi `2im`M T`BK`v rp2}2H/ Bib2H7- BMbi2/ i?2 bQHm@iBQM Bb /2+QKTQb2/ BMiQ i?2 bQm`+2 rp2H2i M/ i?2 mM/2`HvBM; bm`7+2@7`22:`22MǶb 7mM+iBQMX 1bb2MiBHHv- 1SaA +M #2 `2;`/2/ b  K2i?Q/ iQ +?B2p2 T?vbB+HHv@+QMbBbi2Mi UrBi? i?2 bm`7+2 KmHiBTH2bV #HBM/ bQm`+2 bB;Mim`2/2+QMpQHmiBQMX h?Bb bT2+i ?b BMi`B;mBM; BKTHB+iBQMb #Qi? iQ i?2 H;Q`Bi?K+?QB+2 U2X;X- Bi 2M#H2b  rv iQ ++2H2`i2 i?2 +QKTmi2 iBK2 Q7 1SaA r?B+?A BMi`Q/m+2 BM *?Ti2` eV- b r2HH b iQ TQi2MiBH TTHB+iBQMb Q7 i?2 bQHmiBQMXAM T`iB+mH`- KMv b2BbKB+ BM72`2M+2 T`Q+2/m`2b bm+? b p2HQ+Biv iQKQ;`@T?v M/ BK;BM; #2M2}i ;`2iHv 7`QK ?pBM; BM+`2b2/ i2KTQ`H `2bQHmiBQMUrB/2` bT2+i`H #M/rB/i?V BM i?2 /i- M/ ?pBM; ++2bb iQ i?2 T`BK`vrp2}2H/ /2+QMpQHp2/ 7`QK i?2 bQm`+2 rp2H2i ;`2iHv ?2HTb BM i?i `2;`/XAM i?2 M2ti b2+iBQM A rBHH #`B2~v b2;m2 BMiQ /Bb+mbbBM; r?v i?Bb bT2+i Q71SaA Bb BMi2`2biBM; 7`QK  Ki?2KiB+H TQBMi Q7 pB2rX2.5.3 TheEPSI formulationanditsrelationtodeconvolutionAM i?Bb b2+iBQM r2 HQQF i i?2 1SaA T`Q#H2K b  rp2}2H/ /2+QMpQHmiBQMT`Q#H2KX Hi?Qm;? r2 rBHH MQi 7m`i?2` 2tTHQ`2 i?Bb +QM+2Ti 7Q` i?2 `2bi Q7i?2 i?2bBb UbB/2 7`QK iFBM; /pMi;2 Q7 Bi BM +?Ti2` 8V- Bi Bb M2p2`i?2H2bbBMi2`2biBM; iQ K2MiBQM r?v i?2 1SaA T`Q#H2K +M #2 bQHp2/ rBi?Qmi iQQKm+? bbmKTiBQM QM i?2 bQm`+2 bB;Mim`2 qXaQm`+2 bB;Mim`2 /2+QMpQHmiBQM Bb i?2 T`+iB+2 Q7 BMp2`iBM; i?2 +QMpQHm@iBQM KQ/2H 7Q` b2BbKB+ bB;MHbX aT2+B}+HHv- Bi ii2KTib iQ bQHp2 7Q` g BM i?2`2HiBQMp = g ∗ q, UkX8Vr?2`2 p Bb  i`+2 Q7 `2+Q`/2/ b2BbKQ;`K- q Bb i?2 bQm`+2 bB;Mim`2 UivTB+HHvmMFMQrMV- M/ i?2 bvK#QH ∗ /2MQi2b +QMpQHmiBQM BM i?2 iBK2 /QKBMX h?2/2+QMpQHmiBQM Bb T2`7Q`K2/ Qp2` HH TQbbB#H2 bQm`+2 M/ `2+2Bp2` TQbBiBQMbQp2` i?2 bm`p2v2/ bm`7+2- M/ i?2 ;;`2;i2 bQHmiBQM r2 Q#iBM Bb i?2 :`22MǶb7mM+iBQM Q7 i?2 bm#bm`7+2 b2BbKB+ `2bTQMb2Xh?2 +QMpQHmiBQM QT2`iQ` rBi?  ivTB+H b2BbKB+ bQm`+2 bB;Mim`2 ?b H`;2 MmHH bT+2 M/ Bb /B{+mHi iQ BMp2`iX AM ;2M2`H- b2BbKB+ /2+QMpQHmiBQM BbM BHH@TQb2/ T`Q#H2K- rBi? i?2 /B{+mHiv 2b+HiBM; ;`2iHv r?2M i?2 bQm`+2bB;Mim`2 Bb mMFMQrM U+HH2/ #HBM/ /2+QMpQHmiBQMVX PM2 Q7 i?2 }`bi bB;MB}+MiKBH2biQM2b BM i?Bb }2H/ Bb i?2 qB2M2`@G2pBMbQM /2+QMpQHmiBQM /2p2HQT2/ BMi?2 RN8yb U_Q#BMbQM- RN8dV- r?B+? Bb 2bb2MiBHHv  `2;mH`Bx2/ H2bi@b[m`2bK2i?Q/ 7Q` BMp2`iBM;  FMQrM +QMpQHmiBQM QT2`iQ`X hQ/v i?Bb `2KBMbk3*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQMi?2 biTH2 /2+QMpQHmiBQM K2i?Q/ r?2M i?2 bQm`+2 bB;Mim`2 Bb bm{+B2MiHvr2HH@FMQrMXJQbi /2+QMpQHmiBQM T`Q#H2Kb BM b2BbKQHQ;v `2 bQK2r?i T2`7Q`K2/dz#HBM/Ǵ- bBM+2 Bi Bb Q7i2M T`+iB+HHv BKTQbbB#H2 iQ T`2+Bb2Hv /2i2`KBM2 i?22t+i bQm`+2 7mM+iBQM b2Mi BMiQ i?2 mM/2`;`QmM/ 7`QK KM@K/2 2[mBTK2MiX"HBM/ /2+QMpQHmiBQM ?b  Km+? #B;;2` MmHH bT+2 i?M  ivTB+H /2+QMpQ@HmiBQM T`Q#H2K BM i2`Kb Q7 #Qi? g M/ qX b 6B;m`2 kX3 b?Qrb- i?2 #HBM//2+QMpQHmiBQM KQ/2H T`Q/m+2b i?2 2t+i bK2 `2bmHi mM/2` +QKTHBK2Mi`vb+HBM; M/ b?B7iBM;X 6m`i?2`KQ`2-  HBKBi2/ #M/rB/i? BM q HbQ K2Mb i?iBi rBHH ;2M2`i2 i?2 bK2 QmiTmi 2p2M r?2M `M/QK ?B;? M/ HQr 7`2[m2M+vMQBb2 Bb //2/ iQ i?2 KQ/2H 7Q` gXh?2 Hi2 RNdyb iQ i?2 2`Hv RN3yb br  #B; H2T Q7 2zQ`i BM i?2 /2p2HQT@K2Mi Q7 #HBM/ bQm`+2 /2+QMpQHmiBQM mbBM; M2r 2biBKiBQM i?2Q`v iQ Q#iBMi?2 bQm`+2 rp2H2iX h?2 +Hbb Q7 dzKBMBKmK 2Mi`QTvǴ K2i?Q/b /2pBb2/ 2b@iBKiQ`b i?i KtBKBx2b i?2 MQM@:mbbBMBiv Q7 i?2 /2+QMpQHp2/ `2~2+iBp@Biv b2`B2b UqB;;BMb- RNd3c .QMQ?Q- RN3Rc lH`v+? M/ qHF2`- RN3kc pM /2`"M M/ S?K- kyy3c *#`2HHB- RN38c PH/2M#m`;- RN3RVX h?2 B/2 Bb i?i+QMpQHmiBQM Hrvb BM+`2b2b i?2 :mbbBMBiv Q7 i?2 bB;MH- M/ i?2`27Q`2i?2 /2+QMpQHp2/ `2bmHi b?QmH/ #2 b MQM@:mbbBM b TQbbB#H2X Em`iQbBb BbivTB+HHv mb2/ b i?2 K2bm`2 Q7 :mbbBMBiv- i?2`27Q`2 K2i?Q/b Q7 i?Bb ivT2`2 HbQ +HH2/ dz?B;?2`@Q`/2` biiBbiB+bǴ K2i?Q/bX h?2 +QKT`iBp2 MK2Bb K2Mi iQ /BbiBM;mBb? Bib2H7 7`QK 2tBbiBM; b2+QM/@Q`/2` biiBbiB+b K2i?Q/b-bm+? b bT2+i`H r?Bi2MBM;X1p2M i?Qm;? KmHiBTH2 `2KQpH Bb MQi mbmHHv `2;`/2/ b  /2+QMpQHm@iBQM T`Q+2bb- Bi +QK2b Mim`HHv 7`QK i?2 +QM+2Ti Q7 KQ/2HHBM; KmHiBTH2bpB +QMpQHmiBQM Q7 irQ rp2}2H/b b BMi`Q/m+2/ BM i?Bb b2+iBQMX bbmKBM;b #27Q`2 i?i i?2 bm`7+2 `2~2+iBQM QT2`iQ` Bb TT`QtBKi2/ #v −I- i?21SaA KQ/2H U2[miBQM kXjV Q7 i?2 `2+Q`/2/ rp2}2H/ BM i?2 R. +b2 +M #2b22M b  p`BiBQM Q7 i?2 bBKTH2 bQm`+2 bB;Mim`2 /2+QMpQHmiBQM T`Q#H2K Q72[miBQM kX8,p = g ∗ q− g ∗ p UkXeV= g ∗ (q− p).1[miBQM kXe 2bb2MiBHHv +QMiBMb irQ +QMpQHmiBQM `2HiBQMb?BTb, h?2 }`bii2`K g ∗ q- rBi? Qm` bbmKTiBQM Q7  ;HQ#HHv biiBQM`v bQm`+2- Bb bBKTHvi?2 bQm`+2 +QMpQHmiBQM `2HiBQM bii2/ BM 2[miBQM kX8X h?2 b2+QM/ i2`K g∗pT`Q/m+2b i?2 KmHiBTH2b +mb2/ #v i?2 7`22 bm`7+2X AMimBiBp2Hv- i?Bb K2MbkN*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQMx w x ⇤ w10x w/10 x ⇤ wx w x ⇤ w10x w/10 x ⇤ wx w x ⇤ wx(t+ .1) w(t .1) x ⇤ wx w x ⇤ wx+ ⌘ w (x+ ⌘) ⇤ w6B;m`2 kX3 "HBM/ /2+QMpQHmiBQM Bb  /B{+mHi ibFX 6Q`  ;Bp2M Q#b2`piBQMU`B;?i +QHmKMV- M BM}MBi2 MmK#2` Q7 bB;MH x M/ #Hm` F2`M2H w +QK#BMiBQMbKv `2T`Q/m+2 Bi mM/2`  +mbH +QMpQHmiBQM `2HiBQMb?BTX /Ti2/ 7`QK HB7i2/ ℓ1/ℓ2 +QMbi`BMi 7Q` bT`b2 #HBM/ /2+QMpQHmiBQM- 1bb2` 2iX HX- kyR8jy*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM50 100 150 200 250 300 350 400 450 500-0.200.20.40.6original signal (time domain)5 10 15 20 25 30 35-0.500.51blur filter (time domain)50 100 150 200 250 300 350 400 450 500-0.4-0.200.20.40.6original observation (time domain)6B;m`2 kXN  R. 1SaA KQ/2HX h?2 :`22MǶb 7mM+iBQM KQ/2H g Bb THQii2/i i?2 iQT rBi? irQ BKTmHbBp2 2p2MibX h?2 bQm`+2 bB;Mim`2 q Bb  biM/`/_B+F2` rp2H2i THQii2/ BM i?2 KB//H2X h?2 `2bmHiBM; Q#b2`piBQM p T`Q/m+2/i?`Qm;? i?2 1SaA KQ/2H U2[miBQM kXeV Bb THQii2/ i i?2 #QiiQKXi?i KmHiBTH2b `2 T`Q/m+2/ #v mbBM; i?2 `2+Q`/2/ b2BbKB+ `2bTQMb2 UrBi?QTTQbBi2 bB;MV b i?2 bQm`+2 rp2}2H/ BM  M2r b2BbKB+ bm`p2v- `2T`2b2MiBM;bQK2 FBM/ Q7 722/#+F bvbi2KX h?Bb HB;Mb rBi? i?2 MQiBQM i?i bm`7+2 KmHiB@TH2b `2 T`Q/m+2/ #v i?2 b2BbKB+ `2bTQMb2b #QmM+BM; #+F mM/2`;`QmM/ 7i2`?BiiBM; i?2 `2+Q`/2`b i i?2 1`i?Ƕb bm`7+2X aBM+2 +QMpQHmiBQM /Bbi`B#mi2bQp2` //BiBQM- KmHiBTH2 `2KQpH +M i?mb HbQ #2 b22M b /2+QMpQHpBM; i?BbM2r dz7`22@bm`7+2Ǵ bQm`+2 bB;Mim`2 (q− p) 7`QK gXh?Bb FBM/ Q7 722/#+F bvbi2K BMi`Q/m+2b bQK2 p2`v BMi2`2biBM; +QMb2@[m2M+2b iQ i?2 mMB[m2M2bb Q7 i?2 bQHmiBQMb 7Q` kXeX aT2+B}+HHv- i?2 722/#+Fi2`K T`2p2Mib Mv `#Bi``v b+HBM; Q7 q UM/  +QKTH2K2Mi`v b+HBM; Q7gV iQ T`Q/m+2 i?2 bK2 Q#b2`piBQMX 6B;m`2b kXN M/ kXRy /2KQMbi`i2 i?Bb2z2+i- M/ b?Qrb i?i bBKTHv b+HBM; i?2 bQm`+2 bB;Mim`2 q rQmH/ BMi`Q/m+2//BiBQMH bTBF2b BM i?2 KQ/2H g BM Q`/2` iQ T`Q/m+2 i?2 bK2 Q#b2`piBQM pXh?mb b22FBM; i?2 bT`b2bi bQHmiBQM 7Q` g HbQ BKTQb2b //BiBQMH KTHBim/2jR*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQM50 100 150 200 250 300 350 400 450 500-0.100.10.20.3consistently scaled signal (time domain)5 10 15 20 25 30 35-1012scaled blur filter (time domain)50 100 150 200 250 300 350 400 450 500-0.4-0.200.20.40.6observation (time domain)6B;m`2 kXRy h?2 bK2 R. 1SaA KQ/2H b b?QrM BM 6B;m`2 kXN- #mi rBi?i?2 bQm`+2 bB;Mim`2 KQ/2H q UKB//H2 THQiV b+H2/ #v kX q2 +M biBHH }M/ :`22MǶb 7mM+iBQM g UiQT THQiV i?i biBb}2b i?2 Q`B;BMH Q#b2`piBQM p- #mi BiBb H2bb bT`b2 i?M i?2 Q`B;BMH g b b?QrM BM i?2 iQT TM2H Q7 6B;m`2 kXNX+QMbi`BMib UA 2KTB`B+HHv Q#b2`p2  bBKBH` bBimiBQM 7Q` T?b2 BM7Q`KiBQMVQM qX h?Bb FBM/ Q7 `2HiBQM Bb MQM 2tBbi2Mi BM i?2 ivTB+H #HBM/ /2+QMpQHmiBQM-M/ b2`p2b iQ /2KQMbi`i2 r?v  KBMBKmK bT`bBiv +`Bi2`B QM g HQM2 +MT`Q/m+2 ;QQ/ bQHmiBQMb 7Q` i?2 r?QH2 KQ/2H- 2p2M i?Qm;?  H`;2 T`i Q7 i?21SaA T`Q#H2K 2bb2MiBHHv +QMbBbib Q7 #HBM/ /2+QMpQHmiBQMX2.6 Outline of the thesis.m2 iQ i?2 `2HiBp2 MQp2Hiv Q7 1SaA- i?2`2 2tBbi KMv bT2+ib Q7 i?2 K2@i?Q/ i?i- /2bTBi2 Bib BM?2`2Mi /pMi;2b- +QMi`B#mi2b iQ Bib +m``2Mi H+FQ7 rB/2bT`2/ /QTiBQM BM i?2 BM/mbi`vX AM i?Bb i?2bBb A rBHH bi`Bp2 iQ +QM@i`B#mi2 BKT`Qp2K2Mib iQ i?2 1SaA 7Q`KmHiBQM i?i BKT`Qp2 Bib `2HB#BHBiv-+T#BHBiv- M/ T`+iB+HBiv- BM rvb i?i rBHH ?2HT iQ 7m`i?2` i?2 /QTiBQMjk*?Ti2` kX S`BM+BTH2b Q7 bm`7+2 KmHiBTH2 T`2/B+iBQM #v +QMpQHmiBQMQ7 i?Bb K2i?Q/X hQ i?Bb 2M/- i?2 7QHHQrBM; `2b2`+? +?Ti2`b rBHH 7Q+mb QMbQK2 T`iB+mH` bT2+ib Q7 Bib +m``2Mi b?Q`i+QKBM;b- b 7QHHQrb,AM *?Ti2` j- A `27Q`KmHi2 i?2 1SaA T`Q#H2K BMiQ  7Q`K i?i +M#2 KQ`2 2bBHv bQHp2/ #v +QMp2t QTiBKBxiBQM- #v miBHBxBM; M ℓ0 iQ ℓ1 `2Ht@iBQM M/ 2tBbiBM; K+?BM2`B2b BM i?2 H`;2@b+H2 QTiBKBxiBQM +QKKmMBivX.2bTBi2 i?2 mi?Q`b Q7 1SaA `2+Q;MBxBM; i?2 BKTQ`iM+2 Q7  KBMBKmK bT`@bBiv +`Bi2`B- i?2 1SaA K2i?Q/ QMHv `m/BK2Mi`BHv BKTQb2b bT`bBiv #v QMHvmT/iBM; i?2 :`22MǶb 7mM+iBQM KQ/2H rBi? i?2 72r H`;2bi 2H2K2Mib Q7 2+?;`/B2Mi mT/i2X h?Bb /Q2b MQi BM ;2M2`H H2/ iQ i?2 bT`b2bi bQHmiBQM- M/+M T`Q/m+2 KMv ?B;?@7`2[m2M+v 2``Q`b BM i?2 KQ/2H mT/i2X AMbi2/ Q7`2HvBM; QM ?`/ i?`2b?QH/BM; Q7 ;`/B2Mi mT/i2b- A /2p2HQT  #B@+QMp2t QTiB@KBxiBQM 7Q`KmHiBQM 7Q` 1SaA +HH2/ _Q#mbi 1SaAX A rBHH /2KQMbi`i2 i?ii?Bb K2i?Q/ H2/b iQ KQ`2 ++m`i2 bQHmiBQMb 7Q` i?2 :`22MǶb 7mM+iBQM rBi?Km+? H2bb /@?Q+ T`K2i2` ir2FBM;XAM *?Ti2` 9- A BMp2biB;i2  rv iQ BM+Q`TQ`i2  7`22@bm`7+2b+ii2`BM;@#b2/ KQ/2H BMiQ i?2 +QMpQHmiBQM@#b2/ 1SaA 7Q`r`/ KQ/2H Q7i?2 KmHiBTH2bX h?Bb T`Q/m+2b BM/BpB/mH b+ii2`BM; i2`Kb i?i `2+`2i2 i?2bm`7+2 KmHiBTH2b BM/2T2M/2MiHv 7`QK i?2 +QMpQHmiBQM KQ/2HX A b?Qr i?ii?Bb +M #2 mb2/ iQ ++m`i2Hv T`2/B+i i?2 bm`7+2 KmHiBTH2b BMbB/2 Q7 `2br?2`2 i?2`2 r2`2 H`;2 KBbbBM; bKTH2 ;Tb BM i?2 +[mBbBiBQM UM/ BMi2;`@iBQMV bm`7+2 7Q` i?2 /i rp2}2H/- 2bT2+BHHv BM i?2 M2`@Qzb2i `2 r?2`2i?Bb FBM/ Q7 2``Q` Bb i?2 bi`QM;2biX h?Bb FBM/ Q7 TT`Q+? Bb r2H+QK2- bBM+2i?2 +m``2Mi K2i?Q/QHQ;v Bb iQ HbQ 2biBKi2 i?2 KBbbBM; TB2+2b Q7 i?2 /i-r?B+? BM+`2b2 i?2 /M;2` Q7 KmimH Qp2`@}iiBM; #2ir22M i?2 /Bz2`2Mi mM@FMQrM rp2}2H/bX A T`QTQb2 KQ/B}+iBQMb iQ i?2 K2i?Q/b T`QTQb2/ BM i?2T`2pBQmb +?Ti2` i?i `2 M22/2/ BM Q`/2` iQ bm++2bb7mHHv BMp2`i i?Bb m;@K2Mi2/ 7Q`r`/ KQ/2H- M/ b?Qr i?i Bi Bb KQ`2 +T#H2 i?M i?2 +m``2MiK2i?Q/QHQ;v BM 1SaA i KBiB;iBM; M2`@Qzb2i ;TbXAM *?Ti2` 8- A BMi`Q/m+2  KmHiBH2p2H bi`i2;v iQ bB;MB}+MiHv b?Q`i2Mi?2 iBK2 M22/2/ iQ +QKTmi2 bQHmiBQMb iQ i?2 _Q#mbi 1SaA T`Q#H2KX h?BbK2i?Q/ 2tTHQBib i?2 2tBbiBM; ?B;? +QKTmiiBQM +QKTH2tBiv Q7 rp2}2H/ +QM@pQHmiBQM BM `2p2`b2- #v bm;;2biBM; iQ Q|Q/ 2`Hv Bi2`iBQMb Ur?2`2 ++m`+vBb MQi BKTQ`iMiV iQ +Q`b2` bTiBH ;`B/bX GQr@Tbb }Hi2`BM; M/ TQbi@LJPBMi2`TQHiBQM Bb miBHBx2/ iQ `2HB#Hv pQB/ HBbBM; Bbbm2b r?2M KQpBM; #2@ir22M /Bz2`2Mi bKTHBM; ;`B/bX A rBHH MHvx2 i?2 2tT2+i2/ +QKTmiiBQM+QKTH2tBiv `2/m+iBQM- M/ rHF i?`Qm;? i?Bb bi`i2;v mbBM;  bvMi?2iB+ 2t@KTH2 /ib2i- b?QrBM; ?Qr 2bBHv Bi +M #2 BKTH2K2Mi2/ rBi?Qmi +?M;BM;i?2 2tBbiBM; _Q#mbi 1SaA H;Q`Bi?Kb BMi`Q/m+2/ BM i?2 T`2pBQmb +?Ti2`bXjjChapter3Robustestimationofprimariesby ℓ1-normminimizationh?2 1biBKiBQM Q7 S`BK`B2b #v aT`b2 AMp2`bBQM U1SaAV K2i?Q/- BMi`Q/m+2/BM i?2 Hbi +?Ti2`- M22/b iQ BMp2`i 7Q` #Qi? i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQMM/ i?2 bQm`+2 bB;Mim`2 i i?2 bK2 iBK2- b i?2v `2 #Qi? mMFMQrMb BM i?2BMi2;`H 2[miBQM KQ/2H 7Q` bm`7+2 KmHiBTH2bX h?Bb `2bmHib BM M BMp2`bBQMi?i Bb MHQ;Qmb iQ #HBM/ /2+QMpQHmiBQM- rBi?  ?B;? TQi2MiBH 7Q` HQ+H@KBMBK Bbbm2b- r?2`2 bT2+i`H H2F;2 +M Q++m` #2ir22M i?2 irQ mMFMQrMbi?i H2/ iQ BM+Q``2+i bQHmiBQMb 7Q` #Qi?X Ai Bb QTTQb2/ iQ i?2 ivTB+H a_J1T`Q+2/m`2 r?2`2 i?2 bQHmiBQM Q7 i?2 irQ mMFMQrMb `2 BKTHB+BiHv /2+QmTH2/i?`Qm;? i?2 T`2/B+iBQM@bm#i`+iBQM T`Q+2bbX h?2 KBM BMbB;?i Q7 1SaA Bbi?i KQ/2HHBM; i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQM b BKTmHbBp2 bTBF2b TQb2b T`BQ` bi`QM; 2MQm;? iQ Qp2`+QK2 i?2 BM?2`2Mi HQ+H@KBMBK Bbbm2b T`2b2Mi BMi?Bb BMp2`bBQMXAM pM :`Q2M2biBDM M/ o2`b+?mm` Ukyy3c M/ T`iHv BM :mBiiQM M/ o2`@b+?mm`- kyy9V- Bi rb b?QrM i?i i?2 KBMBKBx2` Q7  bT`bBiv K2bm`2 QMi?2 T`BK`v rp2}2H/ vB2H/b i?2 +Q``2+i /2KmHiBTH2/ `2bmHi /m`BM; /TiBp2bm#i`+iBQMX h?2`27Q`2- Bi rQmH/ b22K Mim`H 7Q` 1SaA iQ b22F i?2 bT`b2biTQbbB#H2 bQHmiBQM 7Q` i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQMX >Qr2p2`- b r2 b?QrBM i?Bb +?Ti2`- i?2 1SaA H;Q`Bi?K b T`QTQb2/ BM UpM :`Q2M2biBDM M/ o2`@b+?mm`- kyyNV /Q2b MQi miQKiB+HHv b22F i?2 bT`b2bi bQHmiBQM- M/ H2p2bi?2 #m`/2M Q7 /2i2`KBMBM; i?2 TT`QT`Bi2 H2p2H Q7 bT`bBiv iQ i?2 T`+iB@iBQM2`X 6m`i?2`KQ`2- b A rBHH /2KQMbi`i2- i?2 bT`bBiv +QMi`QH T`K2i2`bBM i?2 Q`B;BMH 1SaA /Q MQi bi`QM;Hv +Q``2Hi2 iQ i?2 }MH bT`bBiv H2p2H Q7j9*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMi?2 bQHmiBQM- bQ 2p2M B7 i?2 bi`m+im`2 Q7 i?2 bm#bm`7+2 Bb T2`72+iHv FMQrMUBX2X- i?2 MmK#2` Q7 `2~2+iQ`bV i?Bb BM7Q`KiBQM +MMQi #2 2z2+iBp2Hv mb2/BM i?2 BMp2`bBQMX h?Bb b?Q`i+QKBM; BKT+ib i?2 2z2+iBp2M2bb Q7 i?2 1SaA BMi2`Kb Q7 Bib ~2tB#BHBiv- `2HB#BHBiv- M/ [mHBiv Q7 Bib bQHmiBQMb- T`iB+mH`Hv BMBib #BHBiv iQ `2+Qp2` BKTQ`iMi Hi2 2p2Mib Q7 HQr KTHBim/2X3.1 Motivationh?2 +2Mi`H Tm`TQb2 Q7 i?Bb +?Ti2` Bb iQ T`QTQb2  M2r TT`Q+? iQ 1SaAi?i /Q2b ii2KTi iQ b22F i?2 bT`b2bi bm`7+2@7`22 :`22MǶb 7mM+iBQM BM +QKTmiiBQMHHv 2{+B2Mi rv- M/ BM i?2 T`Q+2bb 2HBKBMi2 i?2 7`22 T`K@2i2`b BMpQHp2/ BM /2i2`KBMBM; i?2 bQHmiBQM bT`bBivX hQ +?B2p2 i?Bb A /2pBb2 `27Q`KmHiBQM Q7 1SaA b  KQ/B}+iBQM Q7 i?2 #bBb Tm`bmBi QTiBKBxiBQMT`Q#H2K U*?2M M/ .QMQ?Q- RNN9c *?2M 2i HX- kyyRV- M/ /2bB;M  +mbiQKH;Q`Bi?K iQ }M/ Bib bQHmiBQMX h?Bb `2bmHib BM  K2i?Q/ i?i Bb TTHB+#H2iQ  rB/2 p`B2iv Q7 /ib2ib rBi?Qmi i?2 M22/ iQ ir2F BMp2`bBQM T`K2@i2`bX "2+mb2 1SaA Bib2H7 Bb  Km+? KQ`2 +QKTmiiBQMHHv 2tT2MbBp2 K2i?Q/+QKT`2/ iQ 2p2M a_J1- Bib bm++2bb ?BM;2b QM r?2i?2` Bi +M `2HB#Hv ;Bp2K2MBM;7mH `2bmHib rBi? KBMBKH [mHBiv +?2+Fb Qp2`  rB/2 `M;2 Q7 /ib2ibXA 722H i?i 2HBKBMiBQM Q7 7`22 T`K2i2`b M/ i?2 //BiBQMH `Q#mbiM2bb i?i+QK2b rBi? Hrvb bFBM; 7Q` i?2 bT`b2bi bQHmiBQM rBHH #2  +`m+BH 2H2K2Mi7Q` 1SaA iQ #2 ++2Ti2/ b  KQ`2 biM/`/ T`+iB+2 #v BM/mbi`vXA HbQ BK iQ BKT`Qp2 i?2 [mHBiv Q7 1SaAǶb 2biBKi2/ KmHiBTH2@7`22:`22MǶb 7mM+iBQM 7Q` i?Qb2 i?i `2 BMi2`2bi2/ BM mbBM; Bi /B`2+iHv- BMbi2/ Q7Dmbi i?2 T`BK`v rp2}2H/X "2+mb2 A `2;`/ 1SaA b M QTiBKBxiBQM T`Q#@H2K- Qm` `27Q`KmHiBQM iF2b /pMi;2 Q7 `2+2Mi /pM+2b BM H`;2@b+H2bT`b2 `2+Qp2`v i2+?MB[m2b UbT2+B}+HHv i?i Q7 pM /2M "2`; M/ 6`B2/HM@/2`- kyy3- kyRRV 7Q` 7bi2` +QMp2`;2M+2X h?Bb HQM2 `2bmHi2/ BM  MQiB+2#H2BKT`Qp2K2Mi BM i?2 }/2HBiv Q7 i?2 `2+Qp2`2/ bm`7+2@7`22 :`22MǶb 7mM+iBQM+QKT`2/ iQ i?2 Q`B;BMH H;Q`Bi?K mM/2` i?2 bK2 MmK#2` Q7 Bi2`iBQMbX6m`i?2`KQ`2- Qm` 7`K2rQ`F Bb HbQ ~2tB#H2 2MQm;? iQ HHQr 2biBKiBM;i?2 KmHiBTH2@7`22 :`22MǶb 7mM+iBQM BM `#Bi``v i`Mb7Q`K /QKBMb rBi?Qmi+?M;BM; i?2 Qp2`HH H;Q`Bi?KX h?Bb HHQrb i`Mb7Q`K@/QKBM bT`bBiv iQ #2H2p2`;2/ 7Q` 7m`i?2` mTHB7ib BM i?2 [mHBiv Q7 i?2 /B`2+iHv 2biBKi2/ :`22MǶb7mM+iBQMXhQ KQiBpi2 i?2b2 ;QHb- A b?Qr BM 6B;m`2 jXR  #`B27 2tKTH2 Q7 1SaA `2@bmHi Q#iBM2/ #27Q`2 M/ 7i2` Qm` `27Q`KmHiBQM- 7Q+mbBM; QM /B`2+i `2+Qp2`vQ7 bKHH T`BK`v 2p2Mib i Hi2` iBK2bX 6B;m`2 jXR# Bb i?2 i?2 bm`7+2@7`22j8*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM:`22MǶb 7mM+iBQM Q#iBM2/ 7`QK `mMMBM; i?2 Q`B;BMH 1SaA H;Q`Bi?K QM i?2/i b?QrM BM 6B;m`2 jXRX hQ Q#iBM i?Bb `2bmHi A ?/ iQ /2+B/2 QM b2p@2`H i?BM;b, MmK#2` Q7 `2~2+iBQM 2p2Mib HHQr2/ BMiQ i?2 :`22MǶb 7mM+iBQMT2` mT/i2- M 2biBKi2 Q7 i?2 ri2`@#QiiQK KmiBM; iBK2- M/ i?2 iBK2@rBM/Qr rBi?BM r?B+? i?2 :`22MǶb 7mM+iBQM Bb HHQr2/ iQ #2 mT/i2/ BM 2+?Bi2`iBQMX qBi? Kv TT`Q+?- A rb #H2 iQ Q#iBM i?2 bQHmiBQM b?QrM BM6B;m`2 jXR+ rBi?Qmi ?pBM; iQ b2i Mv Q7 i?Qb2 T`K2i2`b- r?BH2 iFBM; i?2bK2 KQmMi Q7 +QKTmiiBQMH iBK2X 6B;m`2 jXR/ /2KQMbi`i2b i?2 TQbbB#H2mTHB7i BM i?2 [mHBiv Q7 i?2 /B`2+i bQHmiBQM #v `2+Qp2`BM; i?2 bQHmiBQM BM ?v#`B/ +m`p2H2i@rp2H2i i`Mb7Q`K /QKBMX h?2 Qp2`HH MmK#2` Q7 mT/i2bQM i?2 :`22MǶb 7mM+iBQM `2KBMb i?2 bK2 BM i?Bb +b2XAi Bb BKTQ`iMi iQ MQi2 #27Q`2?M/ i?i BM i?Bb +?Ti2` A /Q MQi `2;`/QM2 BKTQ`iMi bT2+i Q7 1SaA- r?B+? Bb i?2 bBKmHiM2Qmb `2+QMbi`m+iBQM Q7KBbbBM; M2`@Qzb2i /i i?`Qm;?Qmi i?2 BMp2`bBQMX _i?2`- Qm` b+QT2 Bb HBK@Bi2/ iQ BM@/2Ti? 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BMi?2 T?vbB+H /QKBM- M/ U/V `2bmHib 7`QK Qm` TT`Q+? 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I i?2 B/2MiBivKi`Bt M/ q(ω)  b+H` 7mM+iBQM Q7 7`2[m2M+vX h?2 Ki`Bt R KQ/2Hb  bm`@7+2 `2~2+iBQM QT2`iQ` i?i Bb bbmK2/ iQ #2 −I UBX2X- M B/2H `2~2+iQ`V 7Q`i?2 Tm`TQb2 Q7 i?Bb +?Ti2`X h?2 KBM T?vbB+H `2HiBQMb?BT i?i mM/2`HB2b1SaA Bb i?mb 2tT`2bb2/ bP = GQ+GRP. UjXRVh?2 KBM Tm`TQb2 Q7 1SaA Bb iQ BMp2`i i?2 `2HiBQMb?BT 2tT`2bb2/ BM 2[m@iBQM jXR 7Q` g M/ qX PM i?2 H27i@?M/ bB/2 Q7 2[miBQM jXR Bb P i?2 Q#b2`p2//i- r?BH2 i?2 `B;?i@?M/ bB/2 /2b+`B#2b ?Qr iQ KQ/2H i?Bb /i mbBM; i?2irQ mMFMQrM [mMiBiB2b G M/ Q- mbBM; P b 2bb2MiBHHv M BMi2;`iBQMj3*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMxsxrG(xr,xs, t)6B;m`2 jXk M BHHmbi`iBQM Q7 i?2 `v Ti?b UbQHB/ ;`v HBM2V i`p2HH2/ #vi?2 bm`7+2@7`22 :`22MǶb 7mM+iBQM g(xr, t;xb`+) b /2b+`B#2/ BM i?Bb +?Ti2`XAi BM+Hm/2b HH `2bTQMb2b i TQbBiBQM x /m2 iQ M BKTmHbBp2 Tm`2 pQHmK2BMD2+iBQM bQm`+2 i xb`+ i?i ?p2 MQi #22M `2~2+i2/ #v i?2 7`22 bm`7+2-2t+2Ti 7Q` i?2 bQm`+2 ;?QbiX _vb /`rM BM /b?2/ ;`2v HBM2b `2 MQi T`iQ7 gX Pm` bbmK2/ KQ/2H Q7 i?2 T`BK`v rp2}2H/ Bb  bQm`+2 bB;Mim`27mM+iBQM q Bb BMD2+i2/ i xb`+ M/ `2+Q`/2/ i x r?BH2 7QHHQrBM; i?2 `v Ti?bQ7 gX h?BMF #H+F HBM2 BM/B+i2b i?2 7`22@bm`7+2 U2X;X- ri2`@B` BMi2`7+2Vr?BH2 i?BM #H+F HBM2b BM/B+i2 bm#bm`7+2 `2~2+iQ`bXF2`M2HX h?Bb Ki`Bt 2[miBQM KQ/2Hb  /Bb+`2iBxiBQM Q7 i?2 BM?QKQ;2MQmbBMi2;`H 2[miBQM b?QrM BM UkXkVX .m2 iQ MQBb2- BKT2`72+i mT@/QrM rp2}2H//2+QKTQbBiBQM- M/ BM++m`+B2b BM T?vbB+H bbmKTiBQMb Ubm+? b i?2 T`2b@2M+2 Q7 +`Qbb@HBM2 /BTb r?2M T`Q+2bbBM;  j. HBM2 mbBM;  k. H;Q`Bi?KV- i?2/i rBHH MQi HB2 2MiB`2Hv BM i?2 `M;2 Q7 i?2 KQ/2HX aQHpBM; 2[miBQM jXR 7Q`GM/ Q i?mb `2[mB`2b M BMp2`bBQM i?i KBMBKBx2b i?2 2M2`;v Q7 i?2 KBbKi+?#2ir22M i?2 H27i@?M/ M/ `B;?i@?M/ bB/2b Qp2` HH 7`2[m2M+B2bX AM //BiBQM-i?2 +QK#BM2/ mMFMQrMb G M/ Q 2t+22/ i?2 bBx2 Q7 P M/ `2bmHib BM MmM/2`/2i2`KBM2/ bvbi2K- bQ i?2 BMp2`bBQM `2[mB`2b //BiBQMH +QMbi`BMib QMi?2 bQHmiBQMX JQiBpi2/ #v i?2 BKTmHbBp2 Mim`2 Q7 i?2 :`22MǶb 7mM+iBQM-1SaA mb2b  iBK2 /QKBM bT`bBiv +QMbi`BMi QM g b  `2;mH`BxBM; 7+iQ`XjN*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM3.2.1 EPSI writteninoptimizationformb rb /QM2 BM i?2 T`2pBQmb +?Ti2`- A 2tT`2bb 1SaA b M QTiBKBxiBQMT`Q#H2K BM i?2 7Q`K Q7 KBMBKBxBM; M 2tTHB+Bi Q#D2+iBp2- rBi? i?2 mMFMQrMb`2T`2b2Mi2/ #v p2+iQ` [mMiBiB2bX A ;BM /2}M2 i?2 1SaA 7Q`r`/@KQ/2HHBM;QT2`iQ`M b i?2 `B;?i@?M/ bB/2 Q7 2[miBQM jXR +iBM; Qp2` HH 7`2[m2M+B2bQM G M/ Q M/ ;QBM; mM/2` M BMp2`b2 /Bb+`2i2 6Qm`B2` i`Mb7Q`K BM i?2iBK2 tBb- bm+? i?i r2 +M `2bii2 2[miBQM jXR b p = M(g,q;p)- i?Bb iBK2Qp2` i?2 r?QH2 /ib2i BM T?vbB+H /BK2MbBQMb BMbi2/ Q7 KQMQ+?`QKiB+bHB+2bX Ai Bb KQ`2 +QMp2MB2Mi 7Q` M iQ +i QM mMFMQrMb BM i?2 iBK2 /QKBM#2+mb2 Q7 i?2 bT`bBiv +QMbi`BMib r2 rBHH BKTQb2 BM i?2 iBK2 /QKBMX 6Q`i?2 `2KBM/2` Q7 i?Bb +?Ti2` A rBHH /`QT i?2 2tTHB+Bi T`K2i2`BxiBQM Q7 MQM p- i?mb QMHv r`BiBM; p = M(g,q)X A rBHH `2im`M iQ 2tTHB+BiHv biiBM; i?2T`K2i2`BxiBQM BM i?2 M2ti +?Ti2`- bBM+2 r2 rBHH KF2 mb2 Q7 Bi i?2MXLQi2 i?i B7 r2 }t 2Bi?2` Q7 i?2 `;mK2Mib Q7M(g,q)- Bi #2+QK2b  HBM2`QT2`iQ` rBi? `2bT2+i iQ i?2 `2KBMBM; p`B#H2c i?2`27Q`2 M Bb  #BHBM2`QT2`iQ`X A /2MQi2 i?2 HBM2`BxiBQMb rBi? `2bT2+i iQ g M/ q b Mg˜ M/Mq˜- M/ /2}M2 i?2 T`iBH /2`BpiBp2b ∂M/∂q M/ ∂M/∂g 2pHmi2/ i g˜M/ q˜ UmbBM; i?2 Qp2`?2/ iBH/2 iQ /2MQi2 TT`QtBKiBQMb iQ +imH T?vbB+H2MiBiB2b- Q` 2KTB`B+HHv /2i2`KBM2/ [mMiBiB2bVMq˜g :=(∂M∂g)q˜g = M(g, q˜),Mg˜q :=(∂M∂q)g˜q = M(g˜,q).UjXkV"b2/ QM i?Bb /2b+`BTiBQM- r2 +M mb2 Qm` M2r MQiiBQM iQ 2tT`2bb i?2 1SaAT`Q+2/m`2 b 2[mBpH2Mi iQ bQHpBM; i?2 7QHHQrBM; biM/`/@7Q`K +QMbi`BM2/QTiBKBxiBQM T`Q#H2Kminimizeg,q∈Λ∥p−M(g,q)∥2 bm#D2+i iQ ∥g∥0 ≤ ρ, UjXjVr?2`2 i?2 ℓ0@dzMQ`KǴ ∥g∥0 K2bm`2b i?2 MmK#2` Q7 MQM@x2`Q bKTH2b BM gXS`K2i2` ρ +QMi`QHb i?2 bT`bBiv Q7 i?2 bQHmiBQM 7Q` i?2 bm`7+2@7`22 :`22MǶb7mM+iBQMX h?2 }MH /2KmHiBTH2/ `2bmHi +M #2 Q#iBM2/ BM QM2 Q7 irQ rvb,b  /B`2+i +QMpQHmiBQM Q7 i?2 }MH g˜ M/ q˜ U+HH2/ dz/B`2+i `2bmHiǴV Q` #vbm#i`+iBM; 7`QK p i?2 KmHiBTH2 KQ/2HM(g˜, 0) U+HH2/ dz+QMb2`piBp2 `2bmHiǴVXh?2 Hii2` 2z2+iBp2Hv //b Mv `2bB/m2 rp2}2H/ 7`QK i?2 KBbKi+? i2`K9y*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM#+F BMiQ i?2 bQHmiBQM- M/ +QmH/ #2 /2bB`2/ B7 KTHBim/2 HQbb 7`QK H+F Q7+QMp2`;2M+2 Bb M Bbbm2XA mb2 Λ iQ 2tT`2bb i?2 b2i Q7 +M/B/i2 bQm`+2 bB;Mim`2b q i?i biBb7v Mv/2bB`2/ Q`  T`BQ`B +QMbi`BMib r2 rBb? iQ BKTQb2- 2X;X- /2+v `i2b BM iBK2 M/7`2[m2M+v- b r2HH b T?b2 T`QT2`iB2bX AM i?Bb +?Ti2` A QMHv BKTQb2 i?i qBb +QMbi`BM2/ BM  b?Q`i iBK2 rBM/Qr `QmM/ t = 0- BX2X- q(t) = 0 7Q` |t| ≥ Tr?2M q Bb `2T`2b2Mi2/ BM i?2 iBK2 /QKBMX HbQ- q Bb HHQr2/ iQ #2 MiB@+mbH iQ ++QmMi 7Q` Mv TQbbB#H2 iBK2 b?B7ib BM i?2 `2+Q`/2/ /iX b b?QrMBM pM :`Q2M2biBDM M/ o2`b+?mm` UkyRyV- i?Bb KQ/2H Bb 2bBHv 2ti2M/2/ iQ++QKKQ/i2 KQ`2 +QKTH2t bQm`+2 T`QT2`iB2bX 6Q` 2tKTH2- Bi Bb TQbbB#H2 iQiQ +Tim`2 i?2 Qzb2i /2T2M/2M+2 Q7  /BTQH2 bQm`+2 rp2H2i #v T`K2i2`BxBM;Q b /B;QMH Ki`Bt BMbi2/ Q7  bBM;H2 b+H`X h?Bb rb MQi /QM2 BM Qm`+?Ti2` iQ F22T i?2 `2bmHib /B`2+iHv +QKT`#H2 iQ i?i T`Q/m+2/ #v i?2H;Q`Bi?K /2b+`B#2/ BM pM :`Q2M2biBDM M/ o2`b+?mm` UkyyNVX 6m`i?2`KQ`2-i?2 iBK2 rBM/Qr Bb `2HHv QMHv +`m+BH BM i?2 2`Hv bi;2b Q7 i?2 BMp2`bBQM iQKBiB;i2 Bbbm2b rBi? HQ+H KBMBKX AM T`+iB+2 A }M/ i?i Bi Bb mbmHHv b72 iQ`2KQp2 i?Bb iBK2 rBM/Qr 2MiB`2Hv BM i?2 Hi2` bi;2b Q7 i?2 BMp2`bBQM- r?2Mg Bb bm{+B2MiHv `2+Qp2`2/XAi Bb BKTQ`iMi iQ MQi2 i?i- /m2 iQ i?2 bi`m+im`2 Q7 M(g,q)- Qm` }MH2biBKi2 Q7 q b i?2 T`Q/m+i Q7 M BMp2`bBQM rBHH M2+2bb`BHv +Tim`2 HHT`2T`Q+2bbBM; QM p i?i +M #2 KQ/2HH2/ #v  }Hi2` Q7 i?2 bK2 bi`m+im`2b qX qBi? i?2 bbmKTiBQMb HBbi2/ #Qp2- i?Bb K2Mb i?i M 2biBKi2 Q7q rBHH HbQ +Tim`2 ;HQ#H b+HBM;b- bT2+i`mK b?TBM;- M/ p`BQmb Qi?2`T`2T`Q+2bbBM; i2+?MB[m2b +QKKQMHv TTHB2/ iQ i?2 /iX *QMp2`b2Hv- g Bb+QKTH2i2Hv /2i2`KBM2/ #v i?2 T?vbB+b Q7 i?2 7`22@bm`7+2- M/ Bib KTHBim/2/2T2M/b QMHv QM i?2 `2HiBp2 bi`2M;i? #2ir22M T`BK`v 2p2Mib M/ i?2B`bbQ+Bi2/ KmHiBTH2 `2~2+iBQMb BM i?2 /iX b  +QMb2[m2M+2- i?2 bm++2bbQ7 1SaA rBHH /2T2M/ p2`v Km+? QM i?2 BMi2`MH +QMbBbi2M+v Q7 i?2 `2HiBp2KTHBim/2b #2ir22M T`BK`B2b M/ KmHiBTH2b BM i?2 BMTmi /iXAM i?2B` Q`B;BMH rQ`F QM 1SaA- pM :`Q2M2biBDM M/ o2`b+?mm` UkyyNVT`QTQb2/ iQ TT`QtBKi2Hv bQHp2 jXj #v mT/iBM; 2biBKi2b g˜ M/ q˜ UQp2`@?2/ iBH/2 /2MQi2b 2biBKi2/ [mMiBiB2bV BM M Hi2`MiBM; :mbb@a2B/2H Bi@2`iBQM mbBM; bm++2bbBp2 HBM2`BxiBQMb Q7 M(g,q) b b?QrM BM 2[miBQM jXkXh?2 mT/i2b iQ g˜ M/ q˜ `2 bBKTH2 ;`/B2Mi mT/i2b 7Q` HBM2`Bx2/ T`Q#H2KbXh?Bb b+?2K2 +M #2 /2b+`B#2/ b  +v+HB+ #HQ+F@+QQ`/BMi2 /2b+2Mi H;Q`Bi?K-r?B+? A rBHH #`B2~v /2b+`B#2 BM i?2 M2ti b2+iBQM- #27Q`2 BMp2biB;iBM; bQK2 Q7i?2 TQi2MiBH b?Q`i+QKBM;b Q7 i?2B` Q`B;BMH TT`Q+?X9R*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM3.2.2 OptimizingovermultiplevariablesintheEPSI algorithmh?2 KQbi bi`B;?i7Q`r`/ bi`i2;v 7Q` /2HBM; rBi? KmHiBTH2 mMFMQrMb BbiQ bBKTHv HmKT i?2K iQ;2i?2` BMiQ  bBM;H2 ;;`2;i2 p`B#H2- M/ /2HrBi? i?2 M2r T`Q#H2K mbBM; i?2 bK2 biM/`/ TT`Q+?2b b 7Q` bBM;H2mMFMQrMbX >Qr2p2`- i?Bb TT`Q+? ?b BKK2/Bi2 /B{+mHiB2b r?2M i?2 /B7@72`2Mi mMFMQrMb `2 bi`QM;Hv +QmTH2/ M/ /Bz2` BM b+H2 UuQm M/ Ep2?-RNNeV- bm+? b BM Qm` +b2 Q7 DQBMiHv BMp2`iBM; 7Q` #Qi?  rp2}2H/ M/ BibbQm`+2 bB;Mim`2X q2 HbQ ?p2 +QKTH2i2Hv /Bz2`2Mi +QMbi`BMib 7Q` g M/q- r?B+? 2t+2`#i2b i?2 T`Q#H2K Q7 rQ`FBM; QM i?2K BM mMBbQMX Hi2`M@iBp2Hv- r2 +M /2+`2b2 i?2 Q#D2+iBp2 #v Bi2`iBp2Hv QTiBKBxBM; Qp2` g rBi? q}t2/- i?2M q rBi? g }t2/X h?Bb Bb M 2tKTH2 Q7 i?2 +v+HB+ #HQ+F@+QQ`/BMi2/2b+2Mi K2i?Q/b U"2x/2F 2i HX- RN3dVXTTHvBM; i?Bb TT`Q+? iQ bQHp2 2[miBQM jXj H2/b iQ i?2 Q`B;BMH 1SaAH;Q`Bi?KX q2 +M /2}M2 irQ M2r Q#D2+iBp2 7mM+iBQMb i?i `2 7Q`K2/ #viFBM; i?2 [m/`iB+ Q#D2+iBp2 BM 2[miBQM jXj M/ }tBM; QM2 Q7 i?2 p`B#H2bi bQK2 pHm2 Ub /Bb+mbb2/ #Qp2 i?Bb HbQ HBM2`Bx2b M(g,q) `QmM/ i?ip`B#H2V- r?B+? r2 r`Bi2 b fg˜(q) := 12∥p −Mg˜q∥22 M/ fq˜(g) := 12∥p −Mq˜g∥22X H;Q`Bi?K jXR QmiHBM2b r?i  ivTB+H +v+HB+ +QQ`/BMi2 /2b+2MiTT`Q+? rQmH/ +QM7Q`K iQXH;Q`Bi?K jXR *v+HB+ #HQ+F@+QQ`/BMi2 /2b+2Mi 7Q` 1SaARX `2T2ikX g˜k+1 ← argming fq˜k(g) bm#D2+i iQ ∥g∥0 ≤ ρjX q˜k+1 ← argminq fg˜k+1(q) bm#D2+i iQ q ∈ Λ9X k ← k + 18X mMiBH ∥p−M(g˜k, q˜k)∥2 +QMp2`;2b iQ bQK2 +`Bi2`BA7 r2 /Q MQi BM/BpB/mHHv KBMBKBx2 i?2 irQ T`Q#H2Kb M/ BMbi2/ bBKTHvb22F bQK2 /2+`2b2 BM i?2 Q#D2+iBp2 i 2+? Bi2`iBQM- r2 ``Bp2 i i?2 BM2t+i+v+HB+ #HQ+F@+QQ`/BMi2 /2b+2Mi K2i?Q/X h?2 Q`B;BMH 1SaA H;Q`Bi?K mb2bDmbi  bBM;H2 ;`/B2Mi mT/i2 iQ 7Q`K i?2 bm#b2[m2Mi g˜k+1 M/ q˜k+1 Ur?2`2k Bb i?2 Bi2`iBQM +QmMi2`V- M/ i?2`27Q`2 Bb M 2tKTH2 Q7 bm+?  K2i?Q/XA MQr 2tT`2bb i?2 Q`B;BMH 1SaA H;Q`Bi?K BM i?Bb MQiiBQMX hQ #2;BM- /2b+2Mi /B`2+iBQM 7Q` fq˜0(g) Bb Q#iBM2/ i?Qm;? Bib ;`/B2Mi i g˜ = 0X 6Q` i?Bb}`bi Bi2`iBQM k = 1- B7 r2 bi`i rBi? q˜0 = 0- i?Bb ;`/B2Mi Bb 2[mBpH2Mi iQ i?2KmHiB/BK2MbBQMH miQ+Q``2HiBQM Q7 pX hQ 2Mbm`2 ∥g˜∥0 ≤ ρ i +QKTH2iBQMQ7 i?2 H;Q`Bi?K- i?Bb ;`/B2Mi Bb Tbb2/ iQ bQK2 bT`bB7vBM; QT2`iQ` S i?ix2`Q2b KQbi Q7 Bib 2H2K2MibX h?2 `2bmHiBM; bT`b2 ;`/B2Mi Bb i?2M b+H2/ #v9k*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM HBM2 b2`+? M/ mb2/ iQ 7Q`K g˜1X L2ti- r2 7Q`K i?2 ;`/B2Mi 7Q` fg˜1(q) M/mT/i2 q˜1 BM  bi`B;?i7Q`r`/ 7b?BQM UTQbbB#Hv rBi? M BKTHB+Bi T`QD2+iBQMiQ 2Mbm`2 q˜ ∈ ΛVX h?2 T`Q+2bb i?2M Bi2`i2b- biQTTBM; r?2M  +?Qb2M +`Bi2`BBb biBb}2/- bm+? b M Bi2`iBQM HBKBi Q`  i`;2i KBb}iX H;Q`Bi?K jXkT`QpB/2b  bmKK`v Q7 i?Bb T`Q+2/m`2- rBi? α M/ β /2MQiBM; b+HBM;b i?i2Mbm`2 i?2 Q#D2+iBp2 /2+`2b2b bm{+B2MiHv UBX2X- HBM2@b2`+? b+HBM;bVXH;Q`Bi?K jXk P`B;BMH 1SaA H;Q`Bi?KRX g˜0, q˜0 ← x2`Q p2+iQ`kX k ← 0jX `2T2i9X g˜k+1 ← g˜k + αS(∇fq˜k(gk))8X q˜k+1 ← q˜k + β∇fg˜k+1(qk)eX S`QD2+i q˜k+1 iQ biBb7v q˜k+1 ∈ ΛdX k ← k + 13X mMiBH ∥p−M(g˜k, q˜k)∥2 +QMp2`;2b iQ bQK2 +`Bi2`B- Q` k = kiQiAM Q`/2` iQ BKTQb2 i?2 bT`bBiv +QMbi`BMi M22/2/ QM g˜- HH mT/i2b QMg˜ ;Q2b i?`Qm;?  T`2/2i2`KBM2/ bT`bB7vBM; QT2`iQ` SX >Qr2p2`- bT`b2;`/B2Mi mT/i2b Q7 i?Bb 7Q`K `2 MQi +imHHv ;m`Mi22/ iQ bQHp2 jXj- bQi?2 bQHmiBQM rBHH #2 ?2pBHv BM~m2M+2/ #v i?2 bT2+B}+ +?QB+2 Q7 SX b r2 b22BM i?2 M2ti b2+iBQM- K#B;mBiv BM i?2 +?QB+2 Q7 S Bb HbQ i?2 bQm`+2 Q7 KQbiQ7 i?2 7`22 T`K2i2`b BMpQHp2/ BM i?2 Q`B;BMH 1SaA H;Q`Bi?KX3.2.3 Unpredictablenatureofsparseupdatesh?2 bT2+B}+b Q7 i?2 bT`bB7vBM; QT2`iQ` S ;`2iHv z2+ib i?2 #2?pBQm`Q7 i?2 Q`B;BMH 1SaA H;Q`Bi?KX AM Q`/2` iQ #Qi? biBb7v i?2 i2`KBMiBM;bT`bBiv +QMbi`BMi ∥g˜∥0 ≤ ρ M/ /2+`2b2 i?2 KBb}i Q#D2+iBp2 b Km+? bTQbbB#H2- S b2H2+ib M/ F22Tb i?2 H`;2bi KTHBim/2 ρ/ktot 2H2K2Mib Q7 i?2mT/i2 7Q` g˜- r?2`2 ktot Bb i?2 MmK#2` Q7 iQiH ;`/B2Mi Bi2`iBQMb mb2/ 7Q` i?2T`Q#H2K- r?BH2 i?2 `2KBMBM; 2H2K2Mib `2 b2i iQ x2`QX "2+mb2 ktot Bb ``2HvFMQrM #27Q`2?M/- i?2 T`+iBiBQM2` Bb #m`/2M2/ rBi? i?2 ibF Q7 2biBKiBM;M 2KTB`B+H T2`@mT/i2 bT`bBiv HBKBi ρ˜ BM TH+2 Q7 ρ/ktotX hQ KBiB;i2i?Bb bQK2r?i- 1SaA KF2b //BiBQM bBKTHB}+iBQMb- bm+? b bbmKBM; i?ib2BbKB+ `2~2+iBQM 2p2Mib mbmHHv TT2` BM HH i`+2bX h?Bb HHQrb S iQ #2+``B2/ Qmi BM/2T2M/2MiHv QM  T2` i`+2 #bBb- M/ i?mb ρ˜ +M 2[mBpH2MiHv#2 bT2+B}2/ #v i?2 MmK#2` Q7 H`;2bi@KTHBim/2 2p2Mib F2Ti 7`QK 2+? i`+2X9j*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMPM2 Q7 i?2 +?HH2M;2b Q7 i?2 Q`B;BMH 1SaA H;Q`Bi?K Bb i?i Bi Bb /B{@+mHi iQ DmbiB}#Hv +?QQb2  ρ˜ 7Q` Mv MQM@i`BpBH /ib2iX AMimBiBp2Hv- i?BbT`K2i2` b22Kb iQ +Q``2bTQM/ iQ i?2 MmK#2` Q7 2tT2+i2/ `2~2+iBQM 2p2MibBM i?2 :`22MǶb 7mM+iBQM- M/ rQmH/ i?mb +Q``2Hi2 iQ i?2 2tT2+i2/ MmK#2` Q7bm#bm`7+2 `2~2+iQ`bX >Qr2p2`- #2+mb2 i?2 MmK#2` Q7 ;`/B2Mi mT/i2b iQ`2+Qp2`  T`iB+mH` 2p2Mi rBi? i?2 `B;?i KTHBim/2 +M p`v rBH/Hv- 2p2M BMi?2 KQbi 7pQ`#H2 +b2 BK;BM#H2- r?2`2 i?2 MmK#2` Q7 bm#bm`7+2 `2~2+@iQ`b `2 2MiB`2Hv FMQrM #27Q`2?M/- Bi Bb biBHH /B{+mHi iQ +?QQb2 ρ˜ bm+? i?ii?2 MmK#2` Q7 2p2Mib BM Qm` }MH 2biBKi2 Q7 g˜ Ki+? 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HBM2 BM /2Ti?X h?2 KBM T`i Q7 i?2 H;Q`Bi?K +QM@bBbib Q7 M BMBiBHBxiBQM bi2T iQ TT`QtBKi2 i?2 bQm`+2 bB;Mim`2 UHBM2b k@9V-M/ i?2 KBM Bi2`iBQMb UHBM2b d@R8V Q7 Hi2`MiBM; #2ir22M bQHpBM; ℓ1@MQ`K+QMbi`BM2/ T`Q#H2Kb 7Q` g˜ UHBM2 N@RyV M/ Ki+?BM; 7Q` i?2 bQm`+2 bB;Mim`2q˜ UHBM2 RkVX 6Q` MQr A rBHH H2p2 i?2 i2+?MB+HBiv Q7 i?2 BMBiBHBxiBQM bi2T iQi?2 Hbi T`i Q7 i?Bb b2+iBQM- M/ 7Q+mb BMbi2/ QM ?Qr i?2 KBM HQQT bQHp2bi?2 1SaA T`Q#H2K rBi? i?2 ;QH Q7 Q#iBMBM; i?2 bT`b2bi bQHmiBQM 7Q` gX98*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMH;Q`Bi?K jXj h?2 T`QTQb2/ _Q#mbi 1SaA H;Q`Bi?KRX AMTmi, i`;2i `2bB/mH 2M2`;v σ˜- `2+Q`/2/ bm`7+2 b2BbKB+ /i pkX g˜0 ← BMBiBHBx2 pB bBM;H2 2p2Mi TB+FBM; b /2b+`B#2/ BM i2tijX q˜0 ← argminq ∥p−Mg˜0q∥22 bm#D2+i iQ q ∈ Λ9X g˜0 ← `2b2i iQ x2`Q p2+iQ`8X BMBiBHBx2 Bi2`iBQM +QmMi2` k ← 0- ℓ1@MQ`K +QMbi`BMi τ0 ← 0eX B7 mbBM; i`Mb7Q`Kb- b2i SH b bvMi?2bBb QT2`iQ`- Qi?2`rBb2 b2i SH = IdX `2T2i3X τk+1 ← /2i2`KBM2 7`QK σ˜ M/ τkNX x← argminx ∥p−Mq˜kSHx∥2 bm#D2+i iQ ∥x∥1 ≤ τk+1RyX g˜k+1 ← SHxRRX b+H2 g˜k+1 #v sk+1 ++Q`/BM; iQ UjXdVRkX q˜k+1 ← argminq ∥p−Mg˜k+1q∥2 bm#D2+i iQ q ∈ ΛRjX mM/Q b+HBM; Q7 g˜k+1 #v sk+1R9X k ← k + 1R8X mMiBH ∥p−M(g˜k, q˜k)∥2 ≤ σ˜- Q`  T`2/2i2`KBM2/ Bi2`iBQM HBKBi Bb `2+?2/ReX PmiTmi, 2biBKi2/ :`22MǶb 7mM+iBQM g˜k M/ bQm`+2 rp2H2i q˜k3.3.1 Sparsityviabasispursuit(lines7-15).B`2+iHv TmiiBM; ∥g∥0 BM i?2 Q#D2+iBp2 H2/b iQ  +QK#BMiQ`BH T`Q#H2K-M/ KQbi ?2m`BbiB+b@#b2/ H;Q`Bi?Kb 7Q` T`Q#H2Kb Q7 i?Bb ivT2 `2 mMbi#H2mM/2` MQBb2X AMbi2/- A iF2 i?2 r2HH@FMQrM TT`Q+? 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HQQb2` ℓ1@MQ`K +QMbi`BMi U.QMQ?Q M/ hbB;- kyy3cJHBQmiQp 2i HX- kyy8VXhQ mb2 i?2 +QMiBMmiBQM b+?2K2 BM Qm` "S.L T`Q#H2K- r2 `2TH+2 i?2"S.L T`Q#H2K BM bm#T`Q#H2K jX9 rBi?  ℓ1@+QMbi`BM2/ T`Q#H2K UivTB+HHv+HH2/  GbbQ T`Q#H2KVminimizeg∥p−Mq˜g∥2 bm#D2+i iQ ∥g∥1 ≤ τ˜k, UjXeVr?2`2 τ˜k Bb  ℓ1@MQ`K +QMbi`BMi i?i p`B2b Qp2` i?2 +v+HB+ #HQ+F@+QQ`/BMi2/2b+2Mi- mbBM; k b i?2 Bi2`iBQM +QmMi2`X A7 r2 ?p2  r2HH@/2}M2/ b2`B2bQ7 +QMbi`BMib τ˜0 < τ˜1 < τ˜2 < . . . < τ˜k i?i +QMp2`;2b iQ i?2 KBMBKmK+QMbi`BMi `2[mB`2/ 7Q` i?2 KBb}i 7mM+iBQM iQ `2+?  +?Qb2M 2``Q` H2p2H- r22M/ mT +QMp2`;BM; iQ i?2 bQHmiBQM iQ i?2 "S.L T`Q#H2KX Ai im`Mb Qmi i?ii?2 b2`B2b Q7 +QMbi`BMib τ˜k +M #2 +?Qb2M BM  Mim`H rv #v TTHvBM; L2riQM `QQi@}M/BM; K2i?Q/ iQ i?2 S`2iQ i`/2@Qz +m`p2 Q7 Qm` "S.LT`Q#H2K UpM /2M "2`; M/ 6`B2/HM/2`- kyy3V- b i?2 bQHmiBQM iQ 2+? Q7i?2 GbbQ T`Q#H2Kb ;Bp2 i?2 bHQT2 Q7 i?2 S`2iQ +m`p2 i τ˜k BM +HQb2/ 7Q`KX_2/2`b mM7KBHB` rBi? mbBM; S`2iQ `QQi@}M/BM; 7Q` ℓ1@KBMBKBxiBQM +M`272` iQ TT2M/Bt  r?2`2 Bi Bb 2tTHBM2/ BM /2iBHXh?2 GbbQ T`Q#H2Kb /2b+`B#2/ BM 2[miBQM jXe ?p2  /Bz2`2MiB#H2 Q#D2+@iBp2 7mM+iBQM- M/ `2 2bBHv bQHp2/ UKQbi +QKKQMHv #v T`QD2+i2/ ;`/B2MiK2i?Q/b- b22 pM /2M "2`; M/ 6`B2/HM/2`- kyy3VX .m2 iQ i?2 mb2 Q7 +QMiBM@miBQM- A }M/ i?i i?2 MmK#2` Q7 ;`/B2Mi mT/i2b `2[mB`2/ 7Q` 2+? GbbQT`Q#H2K Bb QM i?2 Q`/2` Q7 RyX h?2 Qp2`HH L2riQM `QQi@}M/BM; K2i?Q/ 7Q`i?2 +Q``2+i τ˜ i?i bQHp2b "S.L- r?2M BMBiBHBx2/ i τ0 = 0 M/  x2`Q bQHm@iBQM p2+iQ`- HbQ i2M/b iQ +QMp2`;2 [mB+FHv- ivTB+HHv rBi?BM 3 iQ Ry L2riQMbi2TbX a22 6B;m`2 jXj 7Q`  ?vTQi?2iB+H bQHmiBQM Ti? Q7 g˜ 7Q` j Bi2`iBQMbQ7 i?2 L2riQM `QQi@}M/BM; K2i?Q/X93*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM˜⌧0 ⌧1 ⌧2 ⌧3solution of Lasso problem(used to improve q)intermediate Lasso solutionsgradient update on gLasso problemkpMq˜g˜k 2⌧ := kg˜k16B;m`2 jXj AHHmbi`iBQM b?QrBM; i?2 2pQHmiBQM Q7 g˜ U;`2v /Qib- #Qi? +B`+H2M/ b[m`2bV BM `2HiBQM iQ i?2 S`2iQ +m`p2 7mM+iBQM b Bi 7QHHQrb  S`2iQ`QQi@}M/BM; +QMiBMmiBQM K2i?Q/ iQr`/b i?2 "S.L bQHmiBQM 7Q`  +?Qb2MKBb}i σ˜X h?2 QTiBKH UKBMBKmKV τ 7Q` i?2 "S.L T`Q#H2K HB2b r?2`2 i?2S`2iQ +m`p2 BMi2`b2+ib i?2 /b?2/ HBM2X aQHmiBQMb iQ i?2 GbbQ T`Q#H2KUb[m`2 /QibV HB2 QM i?2 S`2iQ +m`p2 M/ Bb HbQ i?2 bQHmiBQM iQ M 2[mBpH2Mi"S.L T`Q#H2K- M/ Bb HBF2Hv iQ #2 bT`b2X h?2b2 bQHmiBQMb `2 mb2/ iQ `2}M2i?2 bQm`+2 bB;Mim`2 2biBKi2 UHBM2 Rk Q7 H;Q`Bi?K jXjVXA KF2 i?2 Q#b2`piBQM i?i i?2b2 GbbQ T`Q#H2Kb HbQ b2`p2 b i?2 Mbr2`iQ ?Qr i?2 bQm`+2 bB;Mim`2 b?QmH/ #2 2biBKi2/ i?`Qm;?Qmi i?2 +Qm`b2 Q7i?2 H;Q`Bi?KX b 2+? Q7 i?2 GbbQ T`Q#H2Kb `2 bQHp2/- r2 Q#iBM  b2`B2b Q7TT`QtBKiBQMb iQ i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQM gX .m2 iQ i?2 Mim`2 Q7+QMbi`BM2/ +QMp2t QTiBKBxiBQM T`Q#H2Kb- i?2v `2 HbQ 2+? bQHmiBQMb iQ "S.L T`Q#H2K rBi?  T`iB+mH` /i KBb}i +QMbi`BMi- /2i2`KBM2/ mMB[m2Hv#v τ˜kX q2 i?2`27Q`2 +M DmbiB}#Hv #2HB2p2 i?i 2+? g˜k `2 HbQ bT`b2X6B;m`2 jX9 BHHmbi`i2b i?i i?2 GbbQ T`Q#H2Kb H2/ iQ bT`b2 2biBKi2b Q7 i?2T`BK`v :`22MǶb 7mM+iBQMX h?2b2 `2 i?2 #2bi TQBMib rBi?BM i?2 H;Q`Bi?KiQ `2}M2 Qm` 2biBKiBQM Q7 i?2 bQm`+2 bB;Mim`2 q- b i?2v HHQr q iQ 7mHHv+Tim`2 i?2 bT2+i`H T`QT2`iB2b Q7 i?2 rp2}2H/X "v im`MBM; bm#T`Q#H2K jX9BMiQ  GbbQ T`Q#H2K- r2 2bb2MiBHHv 2M/ mT rBi? M MHQ;m2 Q7 i?2 BM2t+i+v+HB+ #HQ+F@+QQ`/BMi2 /2b+2Mi b+?2K2X9N*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMTrace numberTime (s)20    40    60     80   100    120  140trace numbertime (s)100.250.50.7510.0.UVtrace numbertime (s)20 40 60 80 100 120 14000.250.50.751Trace nu berTime (s)                      0.0.U#V6B;m`2 jX9  ivTB+H 2biBKi2 7Q` i?2 :`22MǶb 7mM+iBQM g˜ i UV i?2 }`bi Bi2`@iBQM Q7  GbbQ T`Q#H2K U2[miBQM jXeV- M/ U#V i i?2 }MH Bi2`iBQMX h?BbGbbQ T`Q#H2K rb bQHp2/ pB  bT2+i`H@T`QD2+i2/ ;`/B2Mi K2i?Q/ UpM /2M"2`; M/ 6`B2/HM/2`- kyy3V mbBM; 3 ;`/B2Mi mT/i2bX h?Bb /2KQMbi`i2bi?i GbbQ T`Q#H2Kb i2M/ iQ `2bmHi BM bT`b2 bQHmiBQMbX3.3.3 Matchingforthesourcewavelet(lines11-13)h?2 rp2H2i Ki+?BM; bi2T Bb `2~2+i2/ BM HBM2 Rk Q7 H;Q`Bi?K jXjX Ai Dmbi BM@pQHp2b bQHpBM; i?2 bm#T`Q#H2K bii2/ BM 2[miBQM jX8- QM+2 i?2 T`2pBQmb GbbQT`Q#H2K Bb bQHp2/ M/  bT`b2 TT`QtBKiBQM Q7 i?2 bm`7+2@7`22 :`22MǶb7mM+iBQM g˜k Bb Q#iBM2/X A /pQ+i2 bQHpBM; i?2 rp2H2i Ki+?BM; T`Q#H2KiQ +QMp2`;2M+2- r?B+? ?TT2Mb [mB+FHv /m2 iQ i?2 H`;2 KQmMi Q7 Q#b2`p2//i +QKT`2/ iQ i?2 bKHH MmK#2` Q7 mMFMQrMb 7Q` i?2 rp2H2iX A7 q BbQTiBKH `QmM/ i?2 +m``2Mi HBM2`BxiBQM bm+? i?i Bib T`iBH /2`BpiBp2 Q7i?2 Q#D2+iBp2 7mM+iBQM pMBb?2b- i?2M i?2 M2ti ;`/B2Mi mT/i2 7Q` g˜ rBHH #2p2`v +HQb2 iQ  ;`/B2Mi 7Q` i?2 Qp2`HH 1SaA T`Q#H2K- MQi Dmbi i?2 GbbQbm#T`Q#H2Kb Ub22 `pFBM M/ pM G22mr2M- kyRk- 7Q` M `;mK2Mi Q7 r?vi?Bb Bb #2M2}+BHVX LQi2 i?i bm#T`Q#H2K jX8 2bb2MiBHHv BMpQHp2b }M/BM; }Hi2` UbiBb7vBM; Mv +QMbi`BMi bT2+B}2/ BM ΛV i?i Ki+?2b irQ rp2}2H/b,8y*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMi?2 +m``2Mi ;m2bb Q7 i?2 :`22MǶb 7mM+iBQM g˜ M/ i?2 +m``2Mi 2biBKi2 Q7 i?2T`BK`v rp2}2H/ p−M(g˜, 0)X Ai Bb pbiHv Qp2`/2i2`KBM2/- M/ B7  bQT?Bb@iB+i2/ H;Q`Bi?K bm+? b G2bi@a[m`2b Z_ USB;2 M/ amM/2`b- RN3kV Bbmb2/- q˜ rBHH +QMp2`;2 2ti`2K2Hv [mB+FHvXh?2 mT/i2/ rp2H2i 2biBKi2 Bb bm#b2[m2MiHv mb2/ iQ 7Q`K  M2r HBM@2`BxiBQM Mq˜k 7Q` i?2 M2ti GbbQ T`Q#H2KX AM i?2Q`v- i?Bb rBHH T2`im`# i?2mM/2`HvBM; S`2iQ +m`p2- M/ r2 +M MQ HQM;2` b72Hv 2tT2+i i?2 Q`B;BMHbQHmiBQM g˜k iQ #2 TT`QtBKi2Hv QTiBKH 7i2` i?2 `2@HBM2`BxiBQMX >Qr2p2`-i?2 }`bi@Q`/2` 2z2+i Q7 i?Bb T2`im`#iBQM H`;2Hv +QMi`B#mi2b iQ BM++m`+B2bBM i?2 +QKTmi2/ τk+1- r?B+? KQbiHv ;Qp2`Mb ?Qr [mB+FHv i?2 `QQi@}M/BM;T`Q+2/m`2 HQ+i2 i?2 KBMBKmK ℓ1@MQ`K τ∗ 7Q` i?2 i`;2i σ˜X h?2 2z2+ib Q72``Q`b BM TT`QtBKiBM; i?2 S`2iQ +m`p2 bQHmiBQM- M/ Bib bm#b2[m2Mi 2z2+iQM i?2 +QMp2`;2M+2 `i2 Q7 i?2 `QQi@}M/BM; T`Q+2/m`2- Bb bim/B2/ BM pM /2M"2`; M/ 6`B2/HM/2` Ukyy3V- r?2`2 i?2 b2MbBiBpBiv Q7 i?2 +QMp2`;2M+2 `i2Q7 τ iQ 2``Q`b BM bKTHBM; i?2 S`2iQ +m`p2 Bb b?QrM iQ #2 /B`2+iHv `2Hi2/ iQi?2 +QM/BiBQMBM; Q7 2+? Mq˜k X 1KTB`B+HHv A 7QmM/ i?i τk+1 b /2i2`KBM2/#27Q`2 M/ 7i2` i?2 `2@HBM2`BxiBQM ``2Hv /Bz2`2/ #v KQ`2 i?M k iQ j T2`@+2Mi- 2bT2+BHHv iQr`/b i?2 2M/ Q7 i?2 L2riQM Bi2`iBQMb- M/ i?2`27Q`2 A`2;`/ i?2 T2`im`#iBQM Q7 i?2 S`2iQ +m`p2 iQ #2 M2;HB;B#H2 QM i?2 [mHBivQ7 i?2 }MH bQHmiBQMbXM BKTQ`iMi i2+?MB+HBiv M22/b iQ #2 //`2bb2/ ?2`2, GbbQ T`Q#H2KbTH+2 M ℓ1@MQ`K +QMbi`BMi QM i?2 :`22MǶb 7mM+iBQMb- r?B+? BM/B`2+iHv HBK@Bib i?2 KTHBim/2 Q7 i?2 T`BK`v `2~2+iBQM 2p2Mib BM g˜kX h?Bb Bb  M2+2bb`vi`/2@Qz mMB[m2 iQ i?2 "S.L TT`Q+? i?i /2T2M/b QM mbBM; i?2 ℓ1@MQ`Kb  ?2m`BbiB+ 7Q` bT`bBivX *QMb2[m2MiBHHv- r2 +M 2tT2+i g˜k iQ mM/2`2biB@Ki2 i?2 KTHBim/2b Q7 i?2 `2~2+iBQM 2p2MibX hQ 2Mbm`2 i?i Qm` 2biBKi2bQ7 q ?p2 i?2 +Q``2+i KTHBim/2- r2 M22/ iQ TT`QT`Bi2Hv `2b+H2 i?2 2p2MibBM g˜kX lM/2` i?2 bbmKTiBQMb Q7  ;HQ#H bQm`+2 bB;Mim`2 7Q` 2p2`v 2p2Mi BMi?2 /i- Bi b?QmH/ bm{+2 iQ b+H2 i?2 2MiB`2 g˜k rBi?  bBM;H2 7+iQ` sk i?iKBMBKBx2b ∥p−Mq˜k−1(skg˜k)∥2X LQi2 i?i i?Bb b+HBM; /Q2b MQi T2`7Q`K i?2dz/2#BbBM;Ǵ i?i Bb +QKKQM BM i?2 bQHmiBQM Q7 "S.L T`Q#H2Kb U7Q` 2tKTH2-6B;m2B`2/Q 2i HX- kyydVX Ai QMHv ii2KTib iQ KBiB;i2 i?2 Bbbm2 i?i GbbQbQHmiBQMb 2`Hv QM BM i?2 L2riQM Bi2`iBQMb `2 Q#iBM2/ rBi?  b2p2`2Hv mM@/2`2biBKi2/ ℓ1@MQ`KX h?Bb Bb 2bb2MiBHHv M 2t+i HBM2@b2`+? b+HBM;- r?B+?Bb ;Bp2M #vsk =g˜Hk MHq˜k−1p∥Mq˜k−1 g˜k∥22, UjXdV8R*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMrBi? i?2 bvK#QH H /2MQiBM; >2`KBiBM /DQBMiX h?Bb b+HBM; M22/b iQ #2`2biQ`2/ #27Q`2 i?2 M2ti GbbQ T`Q#H2K 7Q` g˜k+1 Bb 7Q`KmHi2/- iQ KF2 bm`2r2 `2im`M iQ i?2 M2B;?#Q`?QQ/ Q7 i?2 S`2iQ +m`p2X AM Kv 2tT2`B2M+2 A }M/i?i sk i2M/ iQ bi`i i `QmM/ k iQ j M/ rBHH +QMp2`;2 iQ R b τ +QMp2`;2biQ τ∗- r?B+? bm;;2bib i?i sk KB;?i #2 mb27mH b  +QMp2`;2M+2 K2i`B+ 7Q`i?2 Qp2`HH `QQi@}M/BM; H;Q`Bi?KX h?Bb b+HBM; Bb `2~2+i2/ BM HBM2b RR M/Rj BM H;Q`Bi?K jXjX3.3.4 Incorporatingsparsifyingtransforms(lines6, 9-10)PM2 Q7 i?2 /pMi;2b Q7 i?2 _Q#mbi 1SaA H;Q`Bi?K Bb i?i b22FBM; i?2T`BK`v :`22MǶb 7mM+iBQM g˜ BM i2`Kb Q7 +Q2{+B2Mib Q7 i`Mb7Q`K /QKBMb Bb`2HiBp2Hv bi`B;?i7Q`r`/X h?Bb Bb #2+mb2 i?2 bT`bBiv Q7 Qm` bQHmiBQM +QK2b7`QK bQHpBM; GbbQ T`Q#H2Kb- M/ i?2 H;Q`Bi?Kb r2 miBHBx2 `2 ;MQbiB+ iQi?2 HBM2` 7Q`r`/ QT2`iQ`X A7 i?2 Q`B;BMH 1SaA TT`Q+? Bb mb2/- i?2bT`bB7vBM; QT2`iQ` S- r?B+? Bb 7Q`KmHi2/ rBi? T?vbB+H `;mK2Mib- KB;?iM22/ iQ #2 `2/2bB;M2/ 7Q` i?2 bT2+B}+ i`Mb7Q`K BM  MQM@i`BpBH rv BM Q`/2`iQ `2iBM +QMp2`;2M+2X 6Q` _Q#mbi 1SaA- r2 `2 7`22 iQ +QKTQmM/Mq˜ rBi?Mv HBM2` bvMi?2bBb i`Mb7Q`K rBi?Qmi +?M;BM; i?2 mM/2`HvBM; H;Q`Bi?KX.2T2M/BM; QM i?2 bT2+B}+ i`Mb7Q`K +?Qb2M- i?Bb +M #2 H2p2`;2/ 7Q`b2p2`H #2M2}ibX b /Bb+mbb2/ BM >2``KMM UkyRyV-  bT`b2 `2T`2b2MiiBQM7Q` g˜- bm+? b +m`p2H2ib- rBHH 2M?M+2 i?2 2z2+iBp2M2bb Q7 ℓ1@MQ`K KBMBKBx@iBQM- r?B+? BM im`M bi`2M;i?2Mb Qm` "S.L TT`Q+?X HbQ-  `2T`2b2MiiBQMi?i Bb +QKTQb2/ Q7 iQKb i?i bTMb Qp2` T?vbB+H bT+2 +M #2 +HH2/ mTQMiQ KBiB;i2 bTiBH mM/2`bKTHBM; Bbbm2b BM pX 6Q` 2tKTH2- g˜ +M #2 bQm;?ib +Q2{+B2Mib BM i?2 ?vT2`#QHB+ _/QM /QKBM- MHQ;Qmb iQ i?2 TT`Q+?mb2/ BM pM .2/2K M/ o2`b+?mm` Ukyy8VX A b?Qr BM 6B;m`2 jXR b r2HHb i?2 LmK2`B+H 1tKTH2b b2+iBQM i?i i?Bb +M #2 H2p2`;2/ iQ BKT`Qp2i?2 `2+Qp2`v Q7 r2F Hi2 ``BpHb- r?B+? +M T`Qp2 +?HH2M;BM; 2p2M iQ i?2_Q#mbi 1SaA 7Q`KmHiBQM /m2 iQ +QKT`iBp2Hv r2F2` bbQ+Bi2/ KmHiBTH2b7`QK bT?2`B+H /Bp2`;2M+2 M/  Mim`HHv HQr2` aL_ +QKT`2/ iQ 2`HB2`2p2MibVXhQ KQ/B7v i?2 GbbQ T`Q#H2Kb 7Q` i?Bb 2ti2MbBQM- i?2 bvMi?2bBb QT2`iQ``2T`2b2Mi2/ #v i?2 bvK#QH SH Q7 i?2 i`Mb7Q`K /QKBM- r?B+? KTb i?2/QKBM +Q2{+B2Mib iQ i?2 T?vbB+H bB;MH- bBKTHv M22/b iQ #2 +QKTQmM/2/rBi? Mq˜ BM 2[miBQM jXe iQ 7Q`K  M2r GbbQ T`Q#H2Kminimizex∥p−Mq˜SHx∥2 bm#D2+i iQ ∥x∥1 ≤ τ. UjX3V8k*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMr?B+? Bb bQHp2/ mbBM; 2t+iHv i?2 bK2 H;Q`Bi?K b i?2 QM2 mb2/ 7Q` UjXeVXh?Bb 2bb2MiBHHv `2@T`K2i2`Bx2b i?2 mMFMQrM :`22MǶb 7mM+iBQM b +Q2{@+B2Mib BM i?2 i`Mb7Q`K /QKBMX PM+2 i?2 bQHmiBQM +Q2{+B2Mib x˜k `2 Q#@iBM2/- i?2 T?vbB+H :`22MǶb 7mM+iBQM +M #2 `2+Qp2`2/ QM+2 ;BM #v i?2bvMi?2bBb QT2`iBQM g˜k = SH x˜kX h?Bb b?QmH/ #2 /QM2 #27Q`2 bm#b2[m2MiHv`2}MBM; q˜X "Qi? i?2 GbbQ H;Q`Bi?K M/ i?2 L2riQM Bi2`iBQMb /Q MQi M22/iQ #2 KQ/B}2/- b HQM; b r2 `2/2}M2 A = Mq˜SH BMbi2/ Q7 A = Mq˜ 7Q`2[miBQMb XR M/ XkX h?2b2 //BiBQMb iQ i?2 H;Q`Bi?K iQ ++QKKQ/i2bi?2 i`Mb7Q`K /QKBM Bb `2~2+i2/ BM HBM2b e- N- M/ Ry BM H;Q`Bi?K jXjX3.3.5 Initialcalibrationofsourcesignature(lines2-4)M 2z2+iBp2 TT`Q+? r?2M /2HBM; rBi? T`Q#H2Kb rBi? BM?2`2Mi HQ+H KBM@BK Bbbm2b Bb iQ mb2 b Km+?  T`BQ`B BM7Q`KiBQM b TQbbB#H2 iQ M``Qr /QrMi?2 TT`QT`Bi2 `2;BQM Q7 i?2 bQHmiBQM bT+2 i i?2 Qmib2i Q7 i?2 BMp2`bBQMUb22 T;2 Rje Q7 "B;;b- RNN3- 7Q`  `2H2pMi /Bb+mbbBQMVX 6Q` 2tKTH2- BMi?2 TTHB+iBQM Q7 #HBM/ T?QiQ;`T?B+ /2#Hm``BM;-  +QKKQM bi`i2;v Bb iQ}`bi B/2MiB7v 2/;2b BM i?2 BK;2- r?B+? HHQrb i?2 H;Q`Bi?K iQ [mB+FHv 2b@i#HBb? i?2 +Q``2+i b?T2 Q7 i?2 mMFMQrM TQBMi bT`2/ 7mM+iBQM mbBM; +QMi`bi@KtBKBxBM; Q#D2+iBp2X A }M/ i?i M MHQ;Qmb i2+?MB[m2 Bb bBKB@H`Hv 2z2+iBp2 7Q` Qm` +b2X Ai 2bb2MiBHHv BMpQHp2b +`27mH B/2MiB}+iBQM Q7i?2 H`;2bi BKTmHbBp2 2p2Mi T2` i`+2 BM g˜- r?B+? Bb i?2M Ki+?2/ rBi? i?2/i iQ `TB/Hv Q#iBM  `2bQM#H2 +M/B/i2 7Q` q˜0Xh?Bb BMBiBHBxiBQM bi`ib rBi?  KmHiB/BK2MbBQMH miQ+Q``2HiBQM Q7 pU2[mBpH2Mi iQ  ;`/B2Mi mT/i2 7Q` g r?2M q˜ = 0VX 6`QK 2+? i`+2 Q7i?Bb miQ+Q``2HiBQM- r2 TB+F Qmi i?2 bBM;H2 bi`QM;2bi 2p2Mi i?i +QBM+B/2brBi?  T`BK`v `2~2+iBQM BM p- r?B+? rBHH ivTB+HHv #2 i?2 Q+2M@#QiiQK`2~2+iBQMX h?Bb +M #2 +?B2p2/ BM  p`B2iv Q7 rvb- bm+? b  +QK#BMiBQMQ7 KmiBM; M/ miQKiB+ }`bi #`2F TB+FBM;- M/ +M #2 KMmHHv p2`B}2/ B7M22/2/X q2 i?2M B/2MiB7v Bib T2F HQ+iBQM b i?2 ``BpH iBK2 Q7 M BKTmHbBp2`2~2+iBQM- M/ TH+2  bBM;H2 bTBF2 i i?i iBK2 BM Qm` Qi?2`rBb2 2KTiv BMBiBHKQ/2H Q7 i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQMX 7i2` i?Bb Bb /QM2 7Q` 2+? i`+2-r2 2Mbm`2 i?i i?2 KTHBim/2 Q7 i?2b2 bTBF2b `2 BM i?2 +Q``2+i M2B;?#Q`?QQ/-b+HBM; rBi? i?2 2t+i HBM2@b2`+? 7Q`KmH BM 2[miBQM jXd- b2iiBM;Mq˜k−1g =M0g := M(g, 0)X  7mHH #+F@T`QD2+iBQM +M HbQ #2 /QM2 #v 2ti2M/BM;s iQ  /B;QMH r2B;?iBM;- HHQrBM; 2+? bTBF2 iQ #2 r2B;?i2/ BM/BpB/mHHvXS?vbB+HHv- i?Bb Bb BMi2`T`2i2/ b /DmbiBM; i?2 KTHBim/2b Q7 2p2`v 2p2Mi BMi?2 bm`7+2@7`22 :`22MǶb 7mM+iBQM bQ i?i +QHH2+iBp2Hv i?2v KQ/2H i?2 KmHiBTH22p2Mib b ++m`i2Hv b TQbbB#H2 i?`Qm;? QT2`iQ` M(g˜, 0)X8j*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM6BMHHv- b2iiBM; i?Bb K2i?Q/B+HHv +HB#`i2/ bBM;H2@2p2Mi TT`QtBKiBQMQ7 g b g˜0- r2 +M bQHp2 2[miBQM jX8 iQ [mB+FHv Q#iBM  ;QQ/ BMBiBH bQm`+2bB;Mim`2 2biBKi2 q˜0 rBi? KBMBKH 72` Q7 Qp2`}iiBM;X h?Bb ++QKTHBb?2b Qm`;QH Q7 M2;iBM; i?2 `BbF Q7 }M/BM;  q˜0 bQ rBH/Hv BM+Q``2+i i?i Bi T`2+Hm/2b`2bQM#H2 bQHmiBQMb Q7 g˜ U2p2M r?2M Bi biBb}2b i?2 +QMbi`BMi b2i Λ i?ir2 T`2/2i2`KBM2/ 7Q` qVX PM+2 q˜0 Bb Q#iBM2/- r2 +M QTiBQMHHv +?QQb2iQ KBMBKBx2 i?2 BKT`BMi Q7 i?2 BMBiBHBxiBQM #v /Bb+`/BM; g˜0 M/ `2b2i BiiQ  x2`Q p2+iQ` g˜0 = 0 #27Q`2 bi`iBM; i?2 +v+HB+ #HQ+F@+QQ`/BMi2 /2b+2MiBi2`iBQMbX h?2 bi2Tb /Bb+mbb2/ ?2`2 +Q``2bTQM/ iQ HBM2b k@9 BM H;Q`Bi?K jXjX3.4 Numerical comparison with originalEPSIA /2KQMbi`i2 i?2 T2`7Q`KM+2 Q7 _Q#mbi 1SaA #v TTHvBM; Bi iQ  k. bvM@i?2iB+ b2BbKB+ /ib2i i?i Bb HbQ mb2/ #v pM :`Q2M2biBDM M/ o2`b+?mm`UkyyNVX  b?Qi ;i?2` 7`QK i?2 +2Mi2` Q7 i?2 KQ/2H Bb b?QrM BM 6B;m`2 jX8-rBi? ``Qrb BM/B+iBM; i?2 H`;2bi bm`7+2 KmHiBTH2bX h?2 /ib2i Bb KQ/@2HH2/ 7`QK  k. bvMi?2iB+ KQ/2H i?i rb /2TB+i2/ BM };m`2 R Q7 i?2 7Q`2@K2MiBQM2/ TT2`X Ai ?b  ri2` Hv2` TT`QtBKi2Hv kyy K /22T M/ Hi2`HHv@p`vBM; bHi /QK2 Hv2` i?i Bb bBimi2/ `Qm;?Hv #2ir22M i?2 /2Ti?bQ7 9yy K M/ 3yy KX h?2 bTHBi@bT`2/ T`2bbm`2 /i UmT iQ x2`Q@Qzb2iV BbKQ/2HH2/ #v  b2+QM/@Q`/2` BM iBK2- 7Qm`i?@Q`/2` BM bT+2 }MBi2@/Bz2`2M+2+QmbiB+ T`Q;`K mbBM;  x2`Q@T?b2 _B+F2` rp2H2i rBi? T2F 7`2[m2M+v jy>x b bQm`+2 bB;Mim`2- M/ Bb /2;?Qbi2/ i i?2 `2+2Bp2`@bB/2X q2 ?p2 r2HH@bKTH2/ bQm`+2 HBM2 i?i 2t+iHv +QBM+B/2b rBi? i?2 `2+2Bp2` HBM2- bQ i?2KmHiB/BK2MbBQMH +QMpQHmiBQM 7Q` KmHiBTH2 2biBKiBQM Bb 7`22 Q7 HBbBM; Bbbm2bX 7+iQ` Q7 √ω Bb BM+Hm/2/ BM Mq˜ iQ ++QmMi 7Q` i?2 i2+?MB+H `2[mB`2K2Mii?i g b?QmH/ #2 i?2 MQ`KH /2`BpiBp2 Q7 i?2 :`22MǶb 7mM+iBQM rBi? `2bT2+iiQ i?2 7`22@bm`7+2X h?Bb Ki+?2b i?2 dzQ#HB[mBiv 7+iQ`Ǵ i?i Bb HbQ mb2/ BMpM :`Q2M2biBDM M/ o2`b+?mm` UkyyNV- M/ HHQrb mb iQ +QKT`2 /B`2+iHvrBi? i?2 Q`B;BMH 1SaA `2bmHibX6`QK i?Bb /i A Q#iBM2/ irQ 2biBKi2b Q7 i?2 bm`7+2@7`22 :`22MǶb 7mM+@iBQM ++Q`/BM; iQ i?2 Q`B;BMH 1SaA H;Q`Bi?K b /2b+`B#2/ BM H;Q`Bi?K jXk7i2` 3y Bi2`iBQMbX 6B;m`2 jX8# /2TB+ib i?2 `2bmHi Q#iBM2/ rBi?Qmi i?2 bT`@bB7vBM; QT2`iQ` S UBX2X- F2Ti 2+? ;`/B2Mi mT/i2 7Q` g˜ mMiQm+?2/V- M/b2`p2b iQ BHHmbi`i2 i?2 BKTQ`iM+2 Q7 bT`bBiv BM 1SaAX LQi QMHv rb i?2T`BK`v KmHiBTH2 `2KQpH `2HiBp2Hv BM2z2+imH U2X;X- i?2 ri2`@#QiiQK KmH@89*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMiBTH2 i yX98 bV- #mi HbQ i?2 `2bmHi +H2`Hv /Q2b MQi `2b2K#H2 `2~2+iBQM 2p2Mib/m2 iQ M BKTmHbBp2 bQm`+2X 6B;m`2 jX8+ /2TB+ib i?2 `2bmHi Q#iBM2/ #v b2i@iBM; S iQ F22T i?2 9 H`;2bi KTHBim/2 2p2Mib T2` i`+2 7Q` 2p2`v mT/i2 QMg˜X h?Bb ;`2iHv BKT`Qp2/ i?2 KmHiBTH2 `2D2+iBQM M/ /2HBp2`2/ i?2 BKTmHbBp2TT2`M+2 r2 2tT2+i 7`QK i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQMX"Qi? Q7 i?2b2 `2bmHib `2[mB`2/  i`BM;H2 KmiBM; KbF QM g˜ i?`Qm;?@Qmi i?2 BMp2`bBQM T`Q+2/m`2 Ub T`i Q7 i?2 ;HQ#H iBK2@rBM/Qr BKTHB+BiHvBM+Hm/2/ BM SV iQ T`2p2Mi TH+BM; 2p2Mib #27Q`2 i?2 }`bi ri2`@#QiiQK `2@~2+iBQM 2p2Mi `QmM/ yXk8 bX >Qr2p2`- i?2`2 `2 biBHH pBbB#H2 `iB7+ib i i?2#QmM/`B2b Q7 i?2 i`BM;H2 Kmi2X aBM+2 i?2b2 MiB@+mbH 2p2Mib `2 MQi BMi?2 Q`B;BMH /i- i?2v BM/B+i2 mT/i2 2``Q`b BMi`Q/m+2/ r?2M q˜ Bb b2p2`2HvmM/2`2biBKi2/ BM KTHBim/2- KQbi HBF2Hv i QM2 Q7 i?2 2`HB2` Bi2`iBQMbX"2+mb2 i?2 Q`B;BMH 1SaA BMp2`bBQM /Q2b MQi ii2KTi iQ b22F i?2 bT`b@2bi bQHmiBQM- A }M/ i?i i?2b2 2``Q`b /2+v p2`v bHQrHv rBi? i?2 Bi2`iBQM+QmMiX hQ KF2 i?2b2 `iB7+ib TT2` KQ`2 pBbB#H2 BM i?2 THQib A bHB;?iHv2t;;2`i2/ i?2 +HBTTBM; T2`+2MiBH2 Q7 i?2 +QHQm`KT 7Q` THQib Q7 i?2 :`22MǶb7mM+iBQMb QMHvXJ2Mr?BH2- 6B;m`2 jX8/ b?Qrb i?2 `2bmHi T`Q/m+2/ #v _Q#mbi 1SaAX Ab2i σ˜ iQ kW Q7 iQiH /i 2M2`;v BM MiB+BTiBQM Q7 2/;2 2z2+ib- #mi i?2T`Q;`K Bb i2`KBMi2/ 2`Hv i i?2 2[mBpH2Mi Q7 3k ;`/B2Mi mT/i2b Q7 i?2:`22MǶb 7mM+iBQMX A }M/ i?i 7Q` _Q#mbi 1SaA r2 MQ HQM;2` M22/ iQ /2T2M/QM  iB;?i KmiBM; KbF iQ `2D2+i MiB@+mbH 2p2MibX LQi2 i?2 #b2M+2 Q7 i?27Q`2K2MiBQM2/ MiB@+mbH `iB7+ib- r?B+? BM/B+i2b i?i r2 `2 +HQb2 iQ;QQ/ T?vbB+H TT`QtBKiBQMb rBi? Qm` p˜ M/ q˜X 6m`i?2`KQ`2- r2 +M KQ`2+H2`Hv b22 ?2`2 i?i `2~2+iBQM 2p2Mib b?Qr +?`+i2`BbiB+b Q7 i?2 2tT2+i2//BTQH2 `2bTQMb2 Q7 i?2 :`22MǶb 7mM+iBQM- Hi?Qm;? Bi Bb KBiB;i2/ bQK2r?i #vi?2 BM+HmbBQM Q7 i?2 Q#HB[mBiv 7+iQ`X A b?QmH/ MQi2 i?i KQbi Q7 i?2b2 `iB7+ibTT2` iQ #2 ?B;?@7`2[m2M+v M/ bBi BM i?2 MmHHbT+2 Q7 rp2H2i +QMpQHmiBQMrBi? q- M/ rBHH #2 ?B;?Hv ii2Mmi2/ rBi?  bmBi#H2 HQr@Tbb }Hi2`- #miMQi +QKTH2i2HvX h?Bb +M #2 b22M BM 6B;m`2b jXe M/ jXe# r?B+? b?Qrb i?2:`22MǶb 7mM+iBQM BM 6B;m`2b jX8+ M/ jX8/- `2bT2+iBp2Hv- +QMpQHp2/ rBi? i?22biBKi2/ bQm`+2 bB;Mim`2 q˜ b?QrM BM 6B;m`2 jXdX Ai Bb BMi2`2biBM; iQ }M/i?i _Q#mbi 1SaA /Q2b MQi T`Q/m+2 `iB7+ib Q7 i?Bb Mim`2- M/ b2`p2b iQ/2KQMbi`i2 i?i ℓ1@MQ`K `2;mH`BxiBQM Mim`HHv ii2Mmi2b bKHH `M/QK@HBF2 2p2Mib i?i /Q MQi +QMi`B#mi2 iQ 2tTHBMBM; i?2 /i pB i?2 7Q`r`/KQ/2HX h?Bb Bb QM2 Q7 i?2 `2bQMb r?v Bi Bb Q7i2M mb2/ iQ ?2m`BbiB+HHv Q#iBMi?2 bT`b2bi bQHmiBQMX88*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMbB/2 7`QK i?2 #b2M+2 Q7 `iB7+ib- i?2 }/2HBiv Q7 i?2 :`22MǶb 7mM+iBQMQ#iBM2/ 7`QK i?2 _Q#mbi 1SaA T`Q+2/m`2 Bb HbQ [mHBiiBp2Hv #2ii2`- rBi?i?2 r2F2` `2~2+iBQMb `QmM/ yX3 b M/ i?2 /Bz`+i2/ 2p2Mib /B`2+iHv #2HQri?2 ri2`@#QiiQK `2~2+iBQM KQ`2 +H2MHv /2}M2/ +QKT`2/ iQ i?2 Q`B;BMH1SaA `2bmHibX AM #Qi? T`Q;`Kb i?2 rp2H2i rb HHQr2/ iQ HBp2 rBi?BM ±100 Kb iBK2 rBM/QrX h?2 T`QD2+i2/ ;`/B2Mi K2i?Q/b mb2/ iQ bQHp2 i?2b2i Q7 GbbQ T`Q#H2Kb BM _Q#mbi 1SaA +QMp2`;2/ [mB+FHv- ivTB+HHv rBi?BM 8iQ ky T`QD2+i2/ ;`/B2Mi mT/i2b 7Q` 2+? GbbQ T`Q#H2KX h?2 `QQi@}M/BM;T`Q+2/m`2 mT/i2/ τ  iQiH Q7 e iBK2b i?`Qm;?Qmi i?Bb T`Q+2/m`2X i i?2 2M/Q7 i?2 T`Q+2/m`2- i?2 Q`B;BMH 1SaA +?B2p2/  `2HiBp2 /i@bT+2 ℓ2@MQ`K`2bB/mH Q7 yXydN r?BH2 _Q#mbi 1SaA +?B2p2/  HQr2` yXy99c i?2B` `2bB/m2rp2}2H/b p − M(g˜, q˜) `2 THQii2/ BM 6B;m`2b jXe+ M/ jXe/ `2bT2+iBp2HvXq?BH2 i?2 `2bB/m2 Q7 i?2 Q`B;BMH 1SaA Bb brKT2/ #v i?2 7Q`2K2MiBQM2/`iB7+ib- 7Q` _Q#mbi 1SaA i?2 `2bB/m2 KQ`2 +H2`Hv /2KQMbi`i2b T`ib Q7i?2 /i i?i /Q MQi }i i?2 7Q`r`/ KQ/2H- r?B+? BM i?Bb +b2 b?Qr  Qzb2i@/2T2M/2Mi Mim`2- TQbbB#Hv /m2 iQ 2Bi?2` i?2 Q#HB[mBiv 7+iQ` Q` BKT2`72+i/2;?QbiBM;X b /Bb+mbb2/ BM pM :`Q2M2biBDM M/ o2`b+?mm` UkyyN#V- HQQFBM;i i?2 `2bB/m2 BM i?Bb rv Bb mb27mH 7Q` B/2MiB7vBM; TQi2MiBH BM+QMbBbi2M+B2brBi?BM i?2 /ib2iX3.5 Additional numerical examples3.5.1 SyntheticPluto1.5datasethQ /2KQMbi`i2 i?i i?2 _Q#mbi 1SaA H;Q`Bi?K Bb BMb2MbBiBp2 iQ QTiBKBx@iBQM T`K2i2`b- A TTHv Bi iQ /Bz2`2Mi /ib2ib rBi?Qmi TT`2+B#Hv +?M;@BM; T`K2i2`b 7`QK i?2 pHm2b mb2/ BM i?2 T`2pBQmb b2+iBQMX >2`2 A mb2 i?2SHmiQ RX8 /ib2i Tm#HB+Hv `2H2b2/ #v i?2 aJ_h Co +QMbQ`iBmK- r?B+?rb T`Q/m+2/ mbBM;  7Qm`i?@Q`/2` }MBi2 /Bz2`2M+2 S@ao KQ/2HHBM; +Q/2rBi?  _B+F2` rp2H2i Q7 T2F 7`2[m2M+v R8 >xX LQi QMHv /B/ A F22T i?22t+i bK2 T`K2i2`b mb2/ BM i?2 T`2pBQmb 2tKTH2- A HbQ /B`2+iHv mb2/i?2 T`2bbm`2 `2+Q`/BM; i i?2 bm`7+2- 7Q`2;QBM; Mv T`2T`Q+2bbBM; bm+? b/2;?QbiBM; Q` mT@/QrM rp2}2H/ /2+QKTQbBiBQMX lMHBF2 i?2 2tKTH2 BM i?2T`2pBQmb b2+iBQM- M Q#HB[mBiv 7+iQ` rb MQi BM+Hm/2/ BM i?2 7Q`r`/ KQ/2HXL2p2`i?2H2bb- i?2 BMp2`bBQM T`Q+2bb `2KBM2/ bi#H2 M/ T`Q/m+2/  :`22MǶb7mM+iBQM rBi?  +H2M BKTmHbBp2 TT2`M+2 r?BH2 bm++2bb7mHHv `2D2+iBM; KmH@iBTH2bX a22 6B;m`2jX3# 7Q`  b?Qi ;i?2` THQiX qBi?Qmi Mv +?M;2 BM i?28e*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM0.20.40.60.81.0Time (s)-500 0 500Offset (m)UV0.20.40.60.81.0Time (s)-500 0 500Offset (m)U#V0.20.40.60.81.0Time (s)-500 0 500Offset (m)U+V0.20.40.60.81.0Time (s)-500 0 500Offset (m)U/V6B;m`2 jX8 *QKT`BbQM Q7 i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQM Q#iBM2/ pB/Bz2`2Mi 7Q`KmHiBQMb Q7 1SaAX UV h?2 BMTmi /i bBKmHi2/ 7`QK  k.K`BM2 KQ/2HX ``Qrb BM/B+i2 i?2 H`;2bi bm`7+2 KmHiBTH2bX U#V b?Qrbi?2 `2bmHi T`Q/m+2/ #v i?2 Q`B;BMH 1SaA QmiHBM2/ BM H;Q`Bi?K jXk rBi?QmimbBM; i?2 bT`bB7vBM; QT2`iQ` S QM mT/i2b iQ g˜X q?2M S Bb 2KTHQv2/ QMHH mT/i2b iQ g˜- F22TBM; i?2 9 H`;2bi 2p2Mib T2` i`+2 T2` mT/i2- i?2 `2bmHiQ#iBM2/ Bb b?QrM BM U+VX h?2 `2bmHi T`Q/m+2/ #v _Q#mbi 1SaA QmiHBM2/ BMH;Q`Bi?K jXj Bb b?QrM BM U/V- mbBM;  `Qm;?Hv 2[mBpH2Mi MmK#2` Q7 ;`/B2MimT/i2b +QKT`2/ iQ i?2 irQ Qi?2` bQHmiBQMbX h?2 `2~2+iBQM 2p2Mib BM g b?Qr+?`+i2`BbiB+b Q7 i?2 2tT2+i2/ /BTQH2 `2bTQMb2X  +QKT`BbQM #2ir22M U#VM/ U+V b?Qrb i?i bT`bBiv `2;mH`BxiBQM Bb BKTQ`iMi BM T`Q/m+BM; +Q``2+i2biBKiBQMb Q7 i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQMX h?2 bQHmiBQM T`Q/m+2/ #v_Q#mbi 1SaA Bb +H2M2` M/ KQ`2 7`22 Q7 `iB7+ib +QKT`2/ iQ i?i Q7 i?2Q`B;BMH 1SaA rBi? mb2 Q7 i?2 bT`bB7vBM; QT2`iQ`X8d*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM0.20.40.60.81.0Time (s)-500 0 500Offset (m)UV0.20.40.60.81.0Time (s)-500 0 500Offset (m)U#V0.20.40.60.81.0Time (s)-500 0 500Offset (m)U+V0.20.40.60.81.0Time (s)-500 0 500Offset (m)U/V6B;m`2 jXe h?2 `iB7+ib BMi`Q/m+2/ #v i?2 /@?Q+ bT`bB7vBM; QT2`iQ` BMi?2 Q`B;BMH 1SaA H;Q`Bi?K Ub?QrM BM 6B;m`2 jX8V `2 /BKBMBb?2/ #mi biBHHT`2b2Mi 7i2` +QMpQHmiBQM rBi? i?2 2biBKi2/ bB;Mim`2 q˜- b?QrM BM UV 7Q`i?2 Q`B;BMH 1SaA M/ U#V 7Q` _Q#mbi 1SaAX _2bB/m2 rp2}2H/ Bb b?QrM BMU+V 7Q` i?2 Q`B;BMH 1SaA M/ U/V 7Q` _Q#mbi 1SaAXT`K2i2`b mb2/ +QKT`2/ iQ i?2 T`2pBQmb 2tKTH2- M/ /2bTBi2 i?2 H`;2/Bz2`2M+2 #2ir22M i?2 irQ /ib2ib- _Q#mbi 1SaA rb #H2 iQ T`Q/m+2 +H2M 2biBKi2 Q7 i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQM r?BH2 bm++2bb7mHHv `2D2+i@BM; i?2 KBM bm`7+2 KmHiBTH2b pBbB#H2 i k b M/ jX3 b- BM/B+i2/ #v ``QrbXh?2 H;Q`Bi?K HbQ b?Qr2/ ;QQ/ +QMp2`;2M+2 rBi? +Q?2`2Mi 2M2`;v #`2HvpBbB#H2 BM i?2 iQiH `2bB/m2 p−M(g˜, q˜) b?QrM BM 6B;m`2 jX3/XA HbQ mb2/ i?2 _Q#mbi 1SaA p`BMi i?i b22Fb i?2 bQHmiBQM BM  i`Mb@7Q`K /QKBMX M B/2H +M/B/i2 Bb i?2 j. +m`p2H2i 7`K2 7Q` Bib #BHBiviQ `2T`2b2Mi T`2bi+F rp2}2H/b rBi?  ?B;? /2;`22 Q7 bT`bBivX >Qr2p2`-83*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM−100 0 100−0.500.51Time (ms)Relative amplitude  EPSIRobust EPSIUV10 20 30 40 50 60 70 8000.20.40.60.81Frequency (Hz)Relative amplitude  EPSIRobust EPSIU#V6B;m`2 jXd  +QKT`BbQM #2ir22M i?2 bQm`+2@bB;Mim`2 2biBKi2b T`Q/m+2/#v i?2 Q`B;BMH 1SaA M/ i?2 _Q#mbi 1SaA TT`Q+?2b b UV iBK2 bB;MHb-M/ U#V KTHBim/2 bT2+i`X h?2 Q`B;BMH rp2H2i mb2/ Bb  _B+F2` rp2H2iQ7 T2F 7`2[m2M+v jy >xX "Qi? K2i?Q/b T`Q/m+2 +QKT`#H2 KQ/2Hb Q7 i?2bQm`+2 rp2H2iXA BMbi2/ mb2/  b2T`#H2 i`Mb7Q`K i?i Bb KQ`2 K2M#H2 iQ b+HBM; BM /Bbi`B#mi2/ BKTH2K2MiiBQMX Ai +QMbBbib Q7 k. +m`p2H2ib UrBi? 8 b+H2bM/ Re M;H2bV QM iBK2@bHB+2b Q7 i?2 rp2}2H/ iQ T`QKQi2 +QMiBMmBiv +`QbbbQm`+2@`2+2Bp2` +QQ`/BMi2bX AM i?2 iBK2 /QKBM A +?Qb2 i?2 #bBb bbQ+Bi2/rBi? i?2 /Bb+`22i rp2H2i i`Mb7Q`K U.qhV +QMbi`m+i2/ 7`QK  i?B`/@Q`/2`"iiH2@G2K`Bû rp2H2i U"iiH2- RN3dV- /m2 iQ Bib +QKT+i bmTTQ`i BM iBK2M/ #BHBiv iQ bT`b2Hv `2T`2b2Mi b2BbKB+ iBK2@b2`B2bX LQiiBQMHHv r2 +Mr`Bi2 i?Bb ?v#`B/ `2T`2b2MiiBQM b SH := CH ⊗WH - r?2`2 CH Bb i?2 bvM@i?2bBb QT2`iQ` Q7 i?2 k. +m`p2H2i 7`K2 M/WH Bb i?2 BMp2`b2 .qhX h?Bb?v#`B/ i`Mb7Q`K ?/ #22M bm++2bb7mHHv mb2/ #27Q`2 BM i?2 bT`b2 `2+Qp2`v Q78N*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQMbBKmHiM2QmbHv@+[mB`2/ T`2bi+F b2BbKB+ rp2}2H/b U>2``KMM 2i HX- kyyNVXA mb2/ i?Bb i`Mb7Q`K 7Q` HH i?2 2tKTH2b BM i?2 +?Ti2` BMpQHpBM; i`Mb7Q`K/QKBMbX h?2 iQiH ;`/B2Mi Bi2`iBQM +QmMib r2`2 HbQ F2Ti i?2 bK2 b i?2T?vbB+H /QKBM 2tT2`BK2MibX h?2 `2bmHi +M #2 b22M BM 6B;m`2 jX3+X h?BbK2i?Q/ T`Q/m+2b  bKQQi?2` :`22MǶb 7mM+iBQM rBi? 72r2` +QMiBMmBiv Bbbm2b+`Qbb i`+2b +QKT`2/ iQ i?2 2tKTH2b rBi? bT`bBiv T`QKQiBQM BM i?2 T?vb@B+H /QKBM- #mi bQK2 HQr@7`2[m2M+v `iB7+i BM i?2 +`Qbb@/BT /B`2+iBQM +M#2 b22MX h?Bb Kv #2 //`2bb2/ #v  KQ`2 +`27mH b2H2+iBQM Q7 i?2 i`Mb7Q`Kmb2/- Q` #v b+H2 M/ M;H2@/2T2M/2Mi KbFBM; Q7 i?2 i`Mb7Q`K /QKBM +Q@2{+B2MibX h?Bb `iB7+i Bb MQi T`2b2Mi 7i2` +QMpQHmiBQM rBi? i?2 2biBKi2/bQm`+2 bB;Mim`2 q Ub?QrM BM 6B;m`2 jXNV b Bi Bb QmibB/2 i?2 /QKBMMi b2BbKB+bB;MH #M/XhQ bb2bb i?2 [mHBiv Q7 i?2 KmHiBTH2 `2KQpH +?B2p2/ #v _Q#mbi 1SaA-A b?Qr BM 6B;m`2 jXRy i?2 7mHH LJP bi+F Q7 i?Bb /ib2i mbBM; i?2 T`QpB/2/p2HQ+BiB2bX hQ pQB/ Mv TQi2MiBH Bbbm2 +mb2/ #v KTHBim/2 HQbb iQ i?2`2bB/m2 rp2}2H/- i?2 biM/`/ T`QTQb2/ K2i?Q/ 7Q` 1SaA Bb iQ T`Q/m+2 KTHBim/2 M/ T?b2@+Q``2+i KQ/2H Q7 i?2 bm`7+2 KmHiBTH2 rp2}2H/M(g˜, 0)7`QK i?2 2biBKi2/ :`22MǶb 7mM+iBQM- bQ i?i Bi +M #2 /B`2+iHv bm#i`+i2/7`QK i?2 /iX h?Bb Bb +HH2/ i?2 dz+QMb2`piBp2 T`BK`vǴ `2bmHiX 6B;m`2 jXRy#b?Qrb i?2 +QMb2`piBp2 T`BK`v T`Q/m+2/ #v bm#i`+iBM; 7`QK 6B;m`2 jXRyi?2 bm`7+2 KmHiBTH2 KQ/2H b?QrM BM 6B;m`2 jXRy+- r?B+? Bb +H+mHi2/ 7`QKi?2 :`22MǶb 7mM+iBQM Q#iBM2/ #v _Q#mbi 1SaA BM 6B;m`2 jX3#X h?2 };m`2b`2bmHib b?Qr ;QQ/ b2T`iBQM Q7 T`BK`B2b M/ bm`7+2 KmHiBTH2b 2p2M BM `2br?2`2 i?2 irQ +QKTH2i2Hv Qp2`HT- b BM/B+i2/ #v ``QrbX LQi2 i?i i?Bb/2KmHiBTH2 `2bmHi rb ;BM Q#iBM2/ rBi?Qmi Mv T`K2i2` ir2FBM;X3.5.2 GulfofSuezmarinedatab  }MH /2KQMbi`iBQM Q7 _Q#mbi 1SaAǶb +T#BHBiB2b- A HQQF i  K@`BM2 b2BbKB+ HBM2 7`QK i?2 :mH7 Q7 am2x `2;BQM- rBi?  p2`v b?HHQr Q+2M@#QiiQK +mbBM; bi`QM;Hv `BM;BM; bm`7+2 KmHiBTH2b i?i T2`K2i2b i?2 2MiB`2/ib2iX h?2 T`2@T`Q+2bbBM; BM+Hm/2/  iBK2 /QrM@bKTHBM; 7`QK k Kb iQ 9Kb- kt b?Qi bT+BM; BMi2`TQHiBQM iQ +QBM+B/2 rBi? `2+2Bp2` bT+BM;- _/QM/QKBM M2`@Qzb2i BMi2`TQHiBQM- M/ 2ti2M/BM; pB `2+BT`Q+Biv iQ M2;iBp2Qzb2ib r?2M M22/2/X L2Bi?2` /2;?QbiBM; MQ` mT@/QrM /2+QKTQbBiBQM rbT2`7Q`K2/ QM i?2 /i 2t+2Ti 7Q` M ii2KTi2/ `2KQpH Q7 /B`2+i rp2b #vKmiBM;X h?2 /i +QMiBMb jeR b?Qi `2+Q`/b M/ jeR ?v/`QT?QM2 i`+2b i+Q@HQ+i2/ TQbBiBQMbX _Q#mbi 1SaA rb T2`7Q`K2/ QM i?Bb /i mbBM; #Qi?i?2 T?vbB+H /QKBM UH;Q`Bi?K jXjV M/ i?2 i`Mb7Q`K /QKBM p`BMibXey*?Ti2`jX_Q#mbi2biBKiBQMQ7T`BK`B2b#vℓ1 @MQ`KKBMBKBxiBQM12345Time (s)0 2Offset (km)UV12345Time (s)0 2Offset (km)U#V12345Time (s)0 2Offset (km)U+V12345Time (s)0 2Offset (km)U/V6B;m`2 jX3 a?Qi ;i?2` Q7 SHmiQ RX8 /i Bb b?QrM BM UV- r?BH2 i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQM T`Q/m+2/ #v_Q#mbi 1SaA 7i2` 3y ;`/B2Mi mT/i2b QM i?2 :`22MǶb 7mM+iBQM Bb b?QrM BM U#V 7Q` i?2 T?vbB+H /QKBM bQHmiBQMM/ BM U+V 7Q` i?2 i`Mb7Q`K /QKBM bQHmiBQMX h?2 iQiH `2bB/m2 7`QK 2biBKiBM; i?2 :`22MǶb 7mM+iBQM BM U#VBb b?QrM BM U/VX S`K2i2`b 7Q` i?2 _Q#mbi 1SaA H;Q`Bi?K `2KBM mM+?M;2/ 7`QK i?2 QM2b mb2/ iQ T`Q/m+26B;m`2 jX8/X qBi?Qmi ir2FBM;- H;Q`Bi?K jXj T`Q/m+2/  +H2M bQHmiBQM 7Q` i?2 bm`7+2@7`22 :`22MǶb 7mM+iBQMrBi?Qmi i?2 bm`7+2 KmHiBTH2b i k b M/ jX3 b UBM/B+i2/ #v ``QrbVXeR*?Ti2` jX _Q#mbi 2biBKiBQM Q7 T`BK`B2b #v ℓ1@MQ`K KBMBKBxiBQM−200 0 200−1−0.500.51Time (ms)Relative amplitudeUV10 20 30 40 50 6000.20.40.60.81Frequency (Hz)Relative amplitudeU#V6B;m`2 jXN 1biBKi2 Q7 i?2 bQm`+2 bB;Mim`2 q˜ i?i rb T`Q/m+2/ BM i?2T`Q+2bb Q7 Q#iBMBM; 6B;m`2 jX3# b T`i Q7 _Q#mbi 1SaA- THQii2/ b UV iBK2bB;MHb- M/ U#V KTHBim/2 bT2+i`X h?2 Q`B;BMH rp2H2i mb2/ Bb  _B+F2`rp2H2i Q7 T2F 7`2[m2M+v R8 >xX h?2 2biBKi2/ rp2H2i T`iBHHv +Tim`2/i?2 biiBQM`v T`i Q7 i?2 `2+2Bp2` ;?Qbi i?i rb MQi `2KQp2/ 7`QK i?2 /iT`BQ` iQ _Q#mbi 1SaAXA b?Qr b?Qi ;i?2`b Q7 i?2 :`22MǶb 7mM+iBQM g˜ T`Q/m+2/ #v _Q#mbi 1SaA-rBi? 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M2`@Qzb2i ;TXPM+2 i?2 i`m2 T`BK`v rp2}2H/ b?QrM BM U#V Bb iF2M Qmi- i?2 `2KBMBM;i`m2 bm`7+2 KmHiBTH2b +M #2 +QKT`2/ iQ i?2 BM+Q``2+i KQ/2Hb b?QrM BM};m`2b 9XR# M/ 9XR+X>Qr2p2`- QM i?2 ~BT bB/2- i?2 MQM@HQ+H 2z2+i Q7 i?2 T`2/B+iBQM 2``Q`mHiBKi2Hv 2M#H2b mb iQ T`Q#2 i?2 U+iBQMb Q7 i?2V 7`22 bm`7+2 BMbB/2 i?2/i ;T #b2/ bQH2Hv QM K2bm`2K2Mib QmibB/2 Q7 BiX aBM+2 1SaA BM?2`2MiHv`2HB2b QM rp2}2H/ BMp2`bBQM- i?2`2 `2 2MQm;? /2;`22b Q7 7`22/QK BM i?2T`Q+2bb iQ iF2 /pMi;2 Q7 i?Bb 2z2+iX AM i?2B` Q`B;BMH `2TQ`i QM 1SaA-pM :`Q2M2biBDM M/ o2`b+?mm` UkyyNV T`QTQb2 iQ 2tTHB+BiHv `2+QMbi`m+i p′′7`QK BMi2`K2/Bi2 2biBKi2b Q7 g- bm+? i?i i?2 ++m`+v Q7 M BKT`Qp2b bi?2 H;Q`Bi?K +QMp2`;2bX /TiBM; i?Bb TT`Q+? iQ i?2 _1SaA 7Q`KmHiBQMK2Mb 2ti2M/BM; i?2 /2}MBiBQM Q7Mω rBi? i?2 mMQ#b2`p2/ i`+2b b MQi?2`p`B#H2 BM i?2 7Q`r`/ KQ/2HHBM; QT2`iQ`Mω(G,Q,P′′;P′) := GQ+GR(P′ +P′′), U9XRVdd*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;r?B+? H2/b iQ  KQ`2 +QKTHB+i2/ p2`bBQM Q7 i?2 QTiBKBxiBQM T`Q#H2K UkX9V-rBi? p′′ b M //BiBQMH mMFMQrM,ming, q, p′′∥g∥1 bm#D2+i iQ f(g,q,p′′;p′) ≤ σ, U9XkVf(g,q,p′′;p′) := ∥p′ + p′′ −M(g,q,p′′;p′)∥2. KDQ` Bbbm2 rBi? i?Bb K2i?Q/ Q7 BM/2T2M/2MiHv mT/iBM; g M/ p′′BM  +v+HB+HHv Hi2`MiBM; 7b?BQM Bb i?i r2 2M/ mT M22/BM; iQ BM+Hm/2 i?2BMi2`K2/Bi2 M/ BM+Q``2+i 2biBKi2b Q7 p′′ BMiQ i?2 Q#b2`piBQM i2`K Q7 i?2KBb}i 7mM+iBQMHX Pi?2`rBb2- i?2 T`ib Q7 G BMbB/2 i?2 mMQ#b2`p2/ HQ+iBQMbrBHH i2M/ iQ +QMp2`;2 p2`v bHQrHv- B7 i HH- bBM+2 Bib ;`/B2Mib rQmH/ MQi #2/`Bp2M #v i?2 `2bB/mHb i?i #2HQM; iQ i?2 T`BK`B2b- r?B+? QMHv TT2` BMp′′ BMbB/2 i?2 ;TXh?Bb TT`Q+? HbQ B;MQ`2b i?2 7+i i?i g M/ p′′ `2 iB;?iHv +QmTH2/ BMM Q#pBQmb rvX h?2 bi`QM;2bi T`ib Q7 i?2b2 irQ rp2}2H/b Ui?2 T`BK`B2bVQp2`HT M/ `2Hi2 iQ QM2 MQi?2` i?`Qm;? +QMpQHmiBQM rBi? i?2 rp2H2iX h?27mHH `2HiBQM Bb BKTHB+BiHv /2}M2/ #vP′′ = KcP = Kc[GQ+GRP′ +GRP′′], U9XjVr?2`2 Bi Bb 2pB/2Mi i?i P′′ +M #2 HKQbi +QKTH2i2Hv +?`+i2`Bx2/ #v GXh?2`27Q`2- r2 rQmH/ 2tT2+i i?i i?2B` KmimH T`iBH /2`BpiBp2b ∂p′′/∂gM/ ∂g/∂p′′ b?QmH/ MQi #2 B;MQ`2/ r?2M BM/2T2M/2MiHv mT/iBM; p′′ M/gX q?BH2 ∂p′′/∂g Bb bi`B;?i7Q`r`/ iQ +QKTmi2- i?2 2tT`2bbBQM 7Q` ∂g/∂p′′Bb- +QMp2`b2Hv- [mBi2 i`B+Fv iQ +QKTmi2 bi#Hv bBM+2 Bi M2+2bb`BHv BMpQHp2bQ−1 U/2+QMpQHmiBQM #v i?2 mMFMQrM rp2H2i 7mM+iBQM qVX JQiBpi2/ #vi?Bb Q#b2`piBQM- BM i?Bb +?Ti2` A b22F iQ `2KQp2 p′′ b M BMp2`bBQM p`B#H2HiQ;2i?2`- M/ BMbi2/ ++QmMi 7Q` Bib KmHiBTH2 +QMi`B#miBQM rBi? i2`Kb i?iQMHv BMpQHp2 g- q- M/ i?2 FMQrM /i p′X4.1.2 DeterministiccorrectionofsurfacemultiplepredictionbyscatteringtermsA bi`i Qz 2HBKBMiBM; p′′ 7`QK i?2 1SaA KQ/2HHBM; 7mM+iBQM U2tT`2bbBQM 9XRV#v `2+m`bBp2Hv bm#biBimiBM; Bi rBi? i?2 `B;?i@?M/ bB/2 Q7 2tT`2bbBQM 9XjX h?Bb`2bmHib BM  M2r KQ/2HHBM; 7mM+iBQM M˜ i?i /Q2b MQi /2T2M/ QM p′′- #mi ii?2 +Qbi Q7 ?pBM; BM}MBi2Hv KMv i2`Kb BM  b2`B2b 2tTMbBQMX AM i2`Kb Q7d3*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;KQMQ@7`2[m2M+v /i Ki`B+2b- M˜ Bb r`Bii2M bM˜ω(G,Q;P′) = K[(GQ+GRP′)+ GRKc(GQ+GRP′) U9X9V+ GRKc(GRKc(GQ+GRP′)) U9X8V+ O(G4)]:= K∞∑n=0(GRKc)n (GQ+GRP′) . U9XeVh?2 Qmi2`KQbi KbFBM; QT2`iQ` K Bb M22/2/ iQ Ki+? i?2 /i p′′ QMHv ii?2 Q#b2`p2/ HQ+iBQM- bBM+2 2p2`v i2`K BM i?Bb 2tT`2bbBQM T`Q/m+2b rp2}2H/`2bTQMb2b Qp2` i?2 2MiB`2 /QKBMX LQi2 i?i i?Bb 2tT`2bbBQM Bb MQ HQM;2`HBM2` BM GX h?2 n=1 i2`K U9X9V Bb  b2+QM/@Q`/2` i2`K bBM+2 Bi Bb 2z2+iBp2Hv 7mM+iBQM Q7 G2- M/ bBKBH`Hv i?2 n=2 i2`K U9X8V Bb  i?B`/@Q`/2` i2`KXaBM+2 KmHiBTH2b i?i r2 Q#b2`p2 BM Mim`2 rBHH /2+v BM KTHBim/2 b 7mM+iBQM Q7 i?2B` Q`/2`- r2 +M bbmK2 7`QK T?vbB+H `;mK2Mib i?i i?2bT2+i`H MQ`K UBX2X- H`;2bi bBM;mH` pHm2V Q7 G Bb ∥G∥ < 1- M/ i?2`27Q`2i?2 BM}MBi2@i2`K b2`B2b 2tTMbBQM 9Xe +QMp2`;2b iQKP = K (I−GRKc)−1(GQ+GRP′). U9XdVh?Bb 2tT`2bbBQM bii2b i?i r2 +M `2+Qp2` i?2 Q`B;BMH 1SaA 7Q`r`/ KQ/2HP = GQ+GRP′ +GRP′′ B7 Bi Bb bQK2?Qr TQbbB#H2 iQ BMp2`i i?2 /i KbFK #v bQK2 ?vTQi?2iB+H 2t+i /i BMi2`TQHiBQMXHi?Qm;? 2[miBQM 9Xd Bb MQi T`+iB+HHv mb27mH- Bi /KBib  mb27mH BM@i2`T`2iiBQM Q7 Kv TT`Q+?, i?i Q7  "Q`M b2`B2b bQHmiBQM iQ  GBTTKMM@a+?rBM;2` `2HiBQM #2ir22M i?2 iQiH rp2}2H/ P M/ M dzBM+B/2Mi rp2}2H/Ǵ+QKTQb2/ Q7 i?2 2``QM2Qmb 1SaA T`2/B+iBQM (GQ+GRP′) 7`QK BM+QKTH2i2/iX h?2 T2`im`#iBQM BM i?Bb +b2 Bb i?2 KBbbBM; +QMi`B#miBQM +QKBM; 7`QKrBi?BM i?2 +[mBbBiBQM ;TX h?Bb pB2r KF2b Bi +H2` i?i 2+? i2`K BM 2t@T`2bbBQM 9Xe Bb  i2`K BM i?2 "Q`M b2`B2b 2tTMbBQM /m2 iQ i?2 7`22@bm`7+2`2~2+iQ` rBi?BM i?2 +QM}M2b Q7 i?2 +[mBbBiBQM ;T Ub #QmM/ #v i?2 bi2M+BHKcVX PM2 /B`2+i +QMb2[m2M+2 Q7 i?Bb BMi2`T`2iiBQM Bb i?i r2 +M 2tT2+i HHQ`/2`b Q7 bm`7+2 KmHiBTH2b BM 2p2`v i2`K Q7 2tT`2bbBQM 9Xe- #2+mb2 i?2 BM+B@/2Mi rp2}2H/ Bb MQi QMHv +QKTQb2/ Q7 i?2 T`BK`B2b GQ #mi HbQ T`ib Q7HH bm#b2[m2Mi Q`/2`b Q7 bm`7+2 KmHiBTH2b- b T`Q/m+2/ #v i?2 i2`K GRP′Xh?Bb b+ii2`BM;@#b2/ BMi2`T`2iiBQM Bb T2`?Tb KQbi +QKKQMHv b22M BMHBi2`im`2 QM i?2 BMp2`b2 b+ii2`BM; b2`B2b UAaaV TT`Q+? iQ KmHiBTH2 `2KQpHdN*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;Uq2;H2BM 2i HX- RNNd- kyyjVX >Qr2p2`- mMHBF2 ivTB+H TTHB+iBQMb Q7 AaaK2i?Q/QHQ;v- r?2`2 Bi Bb mb2/ QM bm#bm`7+2 `2~2+iQ`b 7Q` BMi2`MH KmHiBTH2`2KQpH- BM i?Bb +?Ti2` r2 FMQr 2t+iHv i?i i?2 b+ii2`BM; #QmM/`v Bb ii?2 7`22 bm`7+2- M/ ?p2 QMHv `2~2+iBQM 2z2+ib 7`QK i?Bb b+ii2`2`X S2`?TbKQ`2 bB;MB}+MiHv- r2 mb2 i?2 /2`Bp2/ KmHiBTH2b b T`i Q7 i?2 KQ/2HHBM;7mM+iBQM BM i?2 +QMi2ti Q7  7mHH /i@}iiBM; BMp2`bBQM T`Q#H2K 7Q` i?2 T`BK`v2biBKiBQM- BMbi2/ Q7 M /TiBp2 bm#i`+iBQM b+?2K2Xb rBi? HH b2`B2b 2tTMbBQM 2tT`2bbBQMb- Bi Bb BKTQ`iMi iQ BMp2biB;i2?Qr i?2 TT`QtBKiBQM 2``Q` #2?p2b b  7mM+iBQM Q7 i?2 MmK#2` Q7 i2`Kbmb2/X aBM+2 +QKTmiBM; i?2b2 i2`Kb BMpQHp2b T`iBH rp2}2H/ +QMpQHmiBQMbi?i ?p2 bB;MB}+Mi +QKTmiiBQMH +Qbib UHi?Qm;? MQi b Km+? b i?2 7mHHrp2}2H/ +QMpQHmiBQM- b r2 rBHH /Bb+mbb Hi2` BM i?2 +?Ti2`V- Bi Bb T`+iB+HHv/2bB`#H2 iQ HBKBi i?2 MmK#2` Q7 i2`Kb b Km+? b TQbbB#H2X AM i?2 M2tib2+iBQM A +HBK r?i Bb T2`?Tb i?2 KBM BMbB;?i Q7 i?Bb +?Ti2`, i?i p2`v bKHH MmK#2` Q7 i?2b2 i2`Kb- 2p2M Dmbi irQ Q` i?`22- +M #2 2MQm;? iQ++QmMi 7Q` KQbi Q7 i?2 KBbbBM; KmHiBTH2 +QMi`B#miBQMbX4.1.3 Effectsoftermtruncationonaccuracyb /Bb+mbb2/ BM i?2 T`2pBQmb b2+iBQM- i?2 r?QH2 b2`B2b mT iQ i?2 n@i? i2`KQ7 2tT`2bbBQM 9Xe ++QmMib 7Q` HH Q7 i?2 KBbbBM; n@i? Q`/2` KmHiBTH2b +QKBM;7`QK i?2 T`i Q7 i?2 7`22 bm`7+2 +QM}M2/ rBi?BM i?2 +[mBbBiBQM ;TX Ai HbQBM+Hm/2b bQK2 T`i Q7 HH KmHiBTH2b Q7 n+1@bi Q`/2` M/ #Qp2- Hi?Qm;? i?2b2+QMi`B#miBQMb Kv #2 Km+? bKHH2`X 6B;m`2 9Xj b?Qrb i?Bb rBi? b?Qi@;i?2``2T`2b2MiiBQMb Q7 i?2 /Bz2`2Mi i2`Kb BM 2tT`2bbBQM 9Xe 7Q` i?2 bvMi?2iB+/ib2i b?QrM BM 6B;m`2 9XRX q2 +M b22 i?i i?2 KDQ`Biv Q7 i?2b2 i2`KbBb +QKTQb2/ Q7 i?2 n@i? Q`/2` bm`7+2 KmHiBTH2 #v +QKT`BM; i?2 rp27`QMib?T2b M/ i`p2HiBK2b Q7 TM2Hb 9Xj# M/ 9Xj+ iQ i?2 i`m2 T`BK`B2b b?QrMBM 6B;m`2 9Xk#X h?2 ``Qrb BM TM2H 9Xj# M/ 9Xj+ b?Qr irQ TB+2b Q7 i?2KBbbBM; +QMi`B#miBQM Q7 i?2 b2+QM/@Q`/2` bm`7+2 KmHiBTH2- r?B+? r2 +M b227BMiHv BM i?2 n=1 i2`K U2[miBQM 9X9V M/ p2`v T`QKBM2MiHv i?2 n=2 i2`KU2[miBQM 9X8VX h?2 bmK Q7 i?2b2 irQ i2`Kb- b?QrM BM TM2H 9Xj/- T2`72+iHv++QmMib 7Q` i?2 iQiH KBbbBM; +QMi`B#miBQM iQ i?2 rp27`QMi rBi?Qmi `2HvBM;QM Mv /i BMbB/2 i?2 ;TX3y*?Ti2`9XJBiB;iBM;+[mBbBiBQM;TbmbBM;b+ii2`BM;Offset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4UV KmHiBTH2 +QMi`B#m@iBQM 7`QK p′′ U/Bz2`@2M+2 Q7 kXd/ M/ 9XR+VOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4U#V n=1 i2`KOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4U+V n=2 i2`KOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4U/V bmK Q7 # M/ +6B;m`2 9Xj a?Qi@;i?2`b Q7 p`BQmb KmHiBTH2 +QMi`B#miBQM i2`Kb BM i?2 miQ@+QMpQHmiBQM #b2/ 7Q`r`/ KQ/2HHBM;QT2`iQ` M˜(G,Q;P′) Ub?QrM BM 2tT`2bbBQM 9XeV r?2M TTHB2/ iQ i?2 bvMi?2iB+ /ib2i rBi? Ryy K KBbbBM; M2`@Qzb2ib BM p′′ b?QrM BM 6B;m`2 9XRX UV b?Qrb i?2 iQiH 2``Q` BM KmHiBTH2 T`2/B+iBQM /m2 iQ KBbbBM; M2`@Qzb2ibX SM2HbU#V M/ U+V `2 `2bT2+iBp2Hv i?2 }`bi irQ i2`Kb Q7 M˜(G,Q;P′) BMpQHpBM; miQ@+QMpQHmiBQMb rBi? g U2tT`2bbBQMb 9X9M/ 9X8VX U/V b?Qrb i?2 bmK Q7 TM2Hb U#V M/ U+VX *QKT`BM; TM2Hb UV M/ U/V- Bi Bb 2pB/2Mi i?i Dmbi i?2 }`biirQ Q7 i?2 ?B;?2` Q`/2` i2`Kb Q7 2[miBQM 9Xe `2 2MQm;? iQ KQ/2H KQbi Q7 i?2 bB;MB}+Mi 1SaA KmHiBTH2 T`2/B+iBQM2``Q`b /m2 iQ i?2 KBbbBM; /iX h?2 ``Qrb BM/B+i2 i?2 TB+2b Q7 i?2 }`bi M/ b2+QM/ Q`/2` bm`7+2 KmHiBTH2b Q7i?2 Q+2M #QiiQK `2~2+iBQMX3R*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;6B;m`2 9Xj HbQ pBbmHHv /2KQMbi`i2b i?i i?2 bmK Q7 Dmbi i?2 }`bi i?`22i2`Kb Q7 i?2 b2`B2b 2tTMbBQM 9Xe Bb 2MQm;? iQ KQ/2H KQbi Q7 i?2 bB;MB}@+Mi KmHiBTH2 +QMi`B#miBQMb 7`QK i?2 KBbbBM; /i p′′X SM2H 9Xj/- r?B+?b?Qrb i?2 bmK Q7 i?2 n=1 M/ n=2 i2`Kb- 2tTHBMb HH i?2 KBbbBM; +QMi`B@#miBQMb 7`QK p′′ UTM2H 9XjV mT iQ i?2 b2+QM/@Q`/2` bm`7+2 KmHiBTH2- M/HbQ +QMiBMb bQK2 FBM2KiB+ BKT`BMib Q7 HH ?B;?2` Q`/2` QM2bX hQ;2i?2`-i?2b2 irQ i2`Kb r2`2 #H2 iQ #`BM; i?2 MQ`KHBx2/ T`2/B+iBQM 2``Q` U/2}M2/#v ∥pm − M˜(g,q;p′)∥/∥pm∥- r?2`2 pm Bb i?2 i`m2 bm`7+2 KmHiBTH2V /QrM7`QK ≈kkW iQ ≈jWX 6B;m`2 9X9 bmKK`Bx2b i?2 2z2+iBp2M2bb Q7 i?2b2 }`biirQ i2`Kb b  7mM+iBQM Q7 i?2 M2`2bi pBH#H2 Qzb2iX6`QK i?2 ;2M2`H i`2M/ Q7 i?2 +m`p2b BM 6B;m`2 9X9 r2 +M b22 i?i- 7Q` `M;2 Q7 ivTB+H M2`@Qzb2i ;T bBx2b- Dmbi BM+Hm/BM; i?2 n=1 i2`K +M`2/m+2 i?2 MQ`KHBx2/ KmHiBTH2 KQ/2HHBM; 2``Q` Q7 M˜ #v KQ`2 i?M QM2 ?H7U+Q``2bTQM/BM; iQ  e /" BKT`Qp2K2MiV- M/  7m`i?2` ?HpBM; Q7 i?2 2``Q`Bb +?B2p2/ #v BM+Hm/BM; mT iQ i?2 n=2 i2`KX h?2 KQ/2H mb2/ ?2`2 Bb i?2bK2 QM2 b mb2/ #v pM :`Q2M2biBDM M/ o2`b+?mm` UkyyNV- rBi?  ;2MiHv@p`vBM; ri2` +QHmKM /2Ti? `QmM/ kyy K i?`Qm;?Qmi i?2 KQ/2H- M/ ri2`@#QiiQK `2~2+iBpBiv +Q2{+B2Mi Q7 ≈yX9R Ub2~QQ` T@rp2 p2HQ+Biv kXyFKfb- /2MbBiv RX3 ;f+KjVX4.2 Algorithmic considerationsqBi? i?2 M2r m;K2Mi2/ 7Q`r`/@KQ/2HHBM; QT2`iQ` M˜(G,Q;P) b /2}M2/BM 2[miBQM 9Xe mbBM; /i@Ki`Bt MQiiBQM- r2 +M MQr bQHp2  M2r QTiB@KBxiBQM T`Q#H2K #v mbBM; Bi BM TH+2 Q7 i?2 mM+Q``2+i2/ KQ/2HHBM; QT2`iQ`BM i?2 Q`B;BMH _1SaA T`Q#H2K U+X7X 2[miBQM 9XkV,ming, q∥g∥1 bm#D2+i iQ f(g,q;p′) ≤ σ, U9X3Vf(g,q;p′) := ∥p′ − M˜(g,q;p′)∥2.h?2 KBM +?HH2M;2 i?i A rBHH //`2bb BM i?Bb b2+iBQM Bb i?i- +QKT`2/ iQi?2 Q`B;BMH _1SaA T`Q#H2K rBi? 7mHHv@bKTH2/ /i- i?Bb M2r QM2 HQb2b i?2HBM2` `2HiBQMb?BT Q7 i?2 KQ/2HHBM; 7mM+iBQM rBi? gX AM i?2 T`2pBQmb +?Ti2`-r2 `2HB2/ QM i?Bb #BHBM2` T`QT2`iv Q7M(g,q;p′) BM i2`Kb Q7 g M/ q iQ bQHp2T`Q#H2K UkX9V mbBM; M Hi2`MiBM; +QQ`/BMi2@/2b+2Mi K2i?Q/X "2HQr A }`bi#`B2~v `2+QmMi ?Qr i?2 Q`B;BMH _1SaA T`Q#H2K Bb bQHp2/- 7QHHQr2/ #v i?2T`QTQbH Q7 irQ TQbbB#H2 rvb iQ KQ/B7v Qm` bQHmiBQM bi`i2;v iQ ++QmMi3k*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM; 15 180 345 4950.10.20.30.40.50.60.70.8Smallest recorded offset (m)Normalized multiple modeling error  up to n=0 (no correction)up to n=1 (second order)up to n=2 (third order)6B;m`2 9X9 JmHiBTH2 KQ/2HHBM; 2``Q` 7Q` i?2 MQMHBM2` b2`B2b 2tTMbBQM KQ/@2HHBM; QT2`iQ` U2tT`2bbBQM 9XeV b  7mM+iBQM Q7 i2`K i`mM+iBQM M/ i?2Qzb2i ;T BM i?2 /iX h?2 /Bz2`2Mi HBM2b THQi i?Bb MQ`KHBx2/ KQ/2HHBM;2``Q` 7`QK i`mM+iBM; HH i2`Kb 7`QK 2tT`2bbBQM 9Xe rBi? n ?B;?2` i?M i?iBM/B+i2/ BM i?2 H2;2M/X7Q` i?2 //BiBQMH MQMHBM2`Biv /m2 iQ i?2 2tBbi2M+2 Q7 ?B;?2`@Q`/2` i2`Kb BMM˜(G,Q;P′)Xh?2 2tBbiBM; TT`Q+? iQ bQHpBM; i?2 Q`B;BMH _1SaA T`Q#H2K UkX9V BbM Hi2`MiBM; mT/i2 Q7 g M/ qX 1+? M2r Bi2`iBQM Q7 g Bb 7Q`K2/ #v ℓ1@MQ`K +QMbi`BM2/ T`Q#H2K U+HH2/  GbbQ T`Q#H2K- hB#b?B`MB- RNNeV iQ;Bp2  bT`b2 rp27`QMi- r?B+? ?b  #`Q/#M/- BKTmHbBp2 TT2`M+2 BMiBK2X q2 i?2M mb2 Bi iQ 2biBKi2 i?2 rp2H2i q #v Ki+?BM; i?2 BKTmHbBp2rp27`QMib ;BMbi i?2 /iX h?2 MmK#2` Q7 bT`b2 rp27`QMib HHQr2/ BMiQg Bb F2Ti bKHH i i?2 2`Hv Bi2`iBQMb iQ +Tim`2 QMHv i?2 KQbi bB;MB}+MiT`BK`v 2p2Mib- [mB+FHv ;BpBM; mb i?2 #2bi 2biBKi2 7Q` i?2 rp2H2i qX Gi2`Bi2`iBQMb rBHH i?2M ;`/mHHv BM+`2b2 i?2 MmK#2` Q7 2p2Mib HHQr2/ iQ #2ii2`}i i?2 `2KBMBM; /i `2bB/mHXh?Bb `2HtiBQM Q7 bT`bBiv +QMbi`BMi ?TT2Mb BM  i?2Q`2iB+HHv `B;Q`Qmbrv 7Q` _1SaAX h?2 bT`bBiv Q7 g Bb /B`2+iHv BM~m2M+2/ #v i?2 ℓ1@MQ`K+QMbi`BMi pHm2 Q7 i?2 GbbQ T`Q#H2KbX h?2 BMBiBH ℓ1@MQ`K +QMbi`BMib3j*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;bi`i i  bKHH pHm2 i?i Bb /2bB;M2/ iQ #2 M mM/2`2biBKi2 Q7 i?2 i`m2KBMBKmKX Ai Bb i?2M ;`/mHHv BM+`2b2/ i?`Qm;?Qmi i?2 Bi2`iBQMb BM  rvbm+? i?i Bi /Q2b MQi 2t+22/ i?2 ℓ1@MQ`K Q7 i?2 i`m2 bQHmiBQM- v2i TT`Q+?2bBi [mB+FHvX h?Bb b+?2K2 Bb 2M#H2/ #v i?2 +QMp2t T`QT2`iB2b Q7 i?2 S`2iQi`/2@Qz +m`p2 7Q` GbbQ T`Q#H2Kb UpM /2M "2`; M/ 6`B2/HM/2`- kyy3c>2MM2M72Mi 2i HX- kyy3c .m#2+?B2b 2i HX- kyy3VX6Q` i?2 bF2 Q7 2tTQbBiBQM- A MQr `Qm;?Hv QmiHBM2 M/ MmK#2` i?2 KBM+QKTQM2Mib Q7 i?2 _Q#mbi 1SaA H;Q`Bi?K b /2b+`B#2/ BM i?2 T`2pBQmb +?T@i2` UH;Q`Bi?K jXjV- QKBiiBM; bQK2 BMBiBHBxiBQM /2iBHbX q2 }`bi b2i Qm`bQHmiBQM +M/B/i2b p˜0 M/ q˜0 iQ x2`Q p2+iQ`b- M/ i?2 BMp2`bBQM `2bB/m2UT`2/B+iBQM 2``Q`V e iQ pX h?2M r2 Bi2`i2 Qp2` i?2 7QHHQrBM; bi2Tb rBi?Bi2`iBQM +QmMi2` kXRX a2H2+i  M2r ℓ1 MQ`K +QMbi`BMi τk > τk−1- r?B+? Bb 2pHmi2/ i?`Qm;? +HQb2/@7Q`K 2tT`2bbBQM 7`QK i?2 +m``2Mi `2bB/m2 ek- i?2 i`;2i KBb}iσ- M/ i?2 ℓ1@MQ`K Q7 i?2 +m``2Mi bQHmiBQM gkXkX P#iBM gk+1 #v bQHpBM; i?`Qm;?  bT2+i`H T`QD2+i2/ ;`/B2Mi K2i?Q/UpM /2M "2`; M/ 6`B2/HM/2`- kyy3c "B`;BM 2i HX- kyyyV- mbBM; i?2T`2pBQmb bQHmiBQM gk b i?2 BMBiBH ;m2bbming∥p−M(g,qk;p)∥2 bm#D2+i iQ ∥g∥1 < τk. U9XNVjX .Q M 2t+i HBM2@b2`+? b+HBM; QM gk+1 mbBM; i?2 KmHiBTH2 i2`K BMM(g,qk;p) QM p iQ KBMBKBx2 i?2 2z2+i Q7 i?2 ℓ1@MQ`K +QMbi`BMi QMi?2 KTHBim/2 Q7 gk+1X9X P#iBM qk+1 #v bQHpBM;  Ki+?2/@}Hi2`BM; T`Q#H2Kminq∥p−M(gk+1,q;p)∥28X 6Q`K  M2r `2bB/m2 ek+1 = p −M(gk+1,qk+1;p) M/ +?2+F 7Q` +QM@p2`;2M+2 +QM/BiBQMbXeX UA7 2tTHB+BiHv `2+QMbi`m+iBM; /i b BM S`Q#H2K 9XkV aQHp2 7Q` i?2 KBbb@BM; T`i Q7 i?2 /i p′′k+1- mbBM; i?2 T`2pBQmb bQHmiBQM p′′k b i?2BMBiBH ;m2bbminp′′∥p′ + p′′ −M(g,qk;p′ + p′′)∥2M/ 7Q`K M2r /i #v p = p′ + p′′k+1X39*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;6Q` bi2Tb R M/ k r2 /2T2M/ QM i?2 MQiBQM Q7  bKQQi?- KQMQiQMB+S`2iQ +m`p2 UbQK2iBK2b FMQrM b i?2 G@+m`p2V 7Q` T`Q#H2K UkX9V iQ ;Bp2mb b2MbB#H2 ℓ1@MQ`K +QMbi`BMi T`K2i2`b τkX h?2 bKQQi? M/ KQMQiQMB+T`QT2`iv Bb ;2M2`HHv MQ HQM;2` pHB/ r?2M i?2 7Q`r`/ bvbi2K Bb MQMHBM2` BMg- bQ #2HQr A rBHH /Bb+mbb irQ TQbbB#H2 TT`Q+?2b iQ rQ`F `QmM/ i?Bb Bbbm2-M/ T`QTQb2 ?Qr Bi +M #2 TTHB2/ iQ i?2 #Qp2 bi`i2;v iQ /2`Bp2 bmBi#H2K2i?Q/b 7Q` bQHpBM; 9X3X4.2.1 ModifiedGauss-Newtonapproacha2p2`H 2tBbiBM; rQ`Fb QM `2;mH`Bx2/ BMp2`bBQM Q7 miQ@+QMpQHmiBQM 7mM+iBQMb`2Hv QM 2Bi?2` i?2 :mbb@L2riQM K2i?Q/ Q` KQ`2 ;2M2`HHv i?2 G2p2M#2`;@J`[m`/i K2i?Q/ U6H2Bb+?2` 2i HX- RNNNc 6H2Bb+?2` M/ >Q7KMM- RNNeVX AMi?2 bK2 p2BM- A BMi`Q/m+2 M TT`Q+? BMbTB`2/ #v GB 2i HX UkyRkV- r?B+?;Bp2b ?2m`BbiB+HHv bT`b2 bQHmiBQMb iQ i?2 M2r MQMHBM2` _1SaA T`Q#H2K U9X3VmbBM;  KQ/B}2/ :mbb@L2riQM K2i?Q/Xh?2 +`mt Q7 i?Bb K2i?Q/ Bb iQ Hrvb 2Mbm`2 i?i i?2 :`22MǶb 7mM+iBQMmT/i2b ∆g `2 b bT`b2 b TQbbB#H2 7Q`  ;Bp2M KQmMi Q7 /2+`2b2 BMi?2 Q#D2+iBp2X h?Bb Bb +?B2p2/ KQbi 2z2+iBp2Hv #v iFBM; b mT/i2b i?2bQHmiBQM iQ  GbbQ T`Q#H2K 7Q`K2/ `QmM/ i?2 C+Q#BM Jk Q7 M˜ 2pHmi2/i (gk,qk),∆gk+1 = argming∥ek − Jkg∥2 bXiX ∥g∥1 ≤ τk, U9XRyVr?2`2 ek Bb i?2 +m``2Mi MQMHBM2` `2bB/m2 p− M˜(gk,qk)X h?2 ℓ1@MQ`K +QM@bi`BMi τk BM i?Bb +b2 Bb ;BM Q#iBM2/ /2i2`KBMBbiB+HHv #v i?2 QTiBKBxiBQMT`K2i2`b Q7 T`Q#H2K 9XRy- mbBM; i?2 bK2 2tT`2bbBQMb mb2/ BM bi2T RX >Qr@2p2`- bBM+2 r2 `2 2tTHB+BiHv bQHpBM; 7Q` mT/i2b ∆g BMbi2/ Q7 g Bib2H7- i?2T`2pBQmb bQHmiBQM gk UM/ ?2M+2 Bib ℓ1@MQ`KV 7Q` i?Bb HBM2`Bx2/ T`Q#H2K BbHrvb x2`QX 1+? mT/i2 7Q`K2/ i?Bb rv rBHH i?mb i2M/ iQ ?p2  Km+?bKHH2` ℓ1@MQ`K +QMbi`BMi τk +QKT`2/ iQ i?2 }MH ℓ1@MQ`K Q7 i?2 i`m2 bQ@HmiBQM- #mi i?2 bmK Q7 HH i?2 mT/i2b Kv ?p2  }MH MQ`K i?i 2t+22/b BiXh?2 bbmKTiBQM ?2`2 Bb i?i Qm` b2`B2b Q7 bT`b2 mT/i2b rBHH +imHHv bmKBMiQ  bT`b2 }MH bQHmiBQM- r?B+? rBHH /2T2M/ QM i?2 C+Q#BMb 2pQHpBM;bHQrHv 2MQm;? Qp2` i?2 Bi2`iBQMb- r?B+? BKTHB2b i?i i?i HH mT/i2b ∆gb?QmH/ FBM2KiB+HHv HH +QMbBbi Q7 p2`v bBKBH` rp27`QMi b2ibX AM T`+iB+2- A}M/ i?i i?Bb bbmKTiBQM Bb ``2Hv  T`Q#H2K B7 i?2 +[mBbBiBQM ;Tb `2 MQi38*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;iQQ H`;2- M/ i?i r2 ivTB+HHv BM/22/ mbmHHv 2M/ mT rBi? bT`b2 bQHmiBQMb7Q` gXh?2 +iBQM Q7 i?2 C+Q#BM Jk Bb bi`B;?i7Q`r`/ iQ Q#iBM- #mi +imHHv+QKTmiBM; Bi rBHH TQb2  +?HH2M;2X A7 r2 2tTHB+BiHv r`Bi2 Bi Qmi mbBM; i?2 /i@Ki`Bt MQiiBQM- Bi #2+QK2b 2pB/2Mi i?i i?2 +QKTmiiBQMH 2zQ`i `2[mB`2/2b+Hi2b p2`v [mB+FHv rBi? i?2 Q`/2` Q7 i?2 b2`B2b 2tTMbBQM,JkωG = K[(GQk +GRP′)+ GkRKc(GQk +GRP′)+ GRKc(GkQk +GkRP′)+ GkRKc(GkRKc(GQk +GRP′))+ GRKc(GkRKc(GkQk +GkRP′))+ GkRKc(GRKc(GkQk +GkRP′))+ ...].U9XRRVLQi2 i?i i?2 #Qp2 2tT`2bbBQM Bb QMHv r`Bii2M mT iQ i?2 n=2 MQMHBM2`i2`K U2tT`2bbBQM 9X8V Q7 M˜ X *QKT`BM; HH i?2 i2`Kb ?2`2 iQ i?2 Q`B;BMHMQMHBM2` 2tT`2bbBQM 9Xe- r2 +M b22 i?i /m2 iQ T`iBH /2`BpiBp2b- 2+?Q7 i?2 O(Gn) i2`Kb 2tTM/ BMiQ n b2T`i2 i2`Kb BM i?2 C+Q#BMX AM;;`2;i2- i?Bb 2z2+iBp2Hv BMi`Q/m+2b  [m/`iB+ +QKTH2tBiv 7+iQ` BM i?2MmK#2` Q7 MQMHBM2` i2`Kb n Qp2` i?2 mbmH +Qbi Q7 i?2 ;`/B2MiX Hi?Qm;?bQK2 `2mb2 Bb TQbbB#H2 BM i?2 +H+mHiBQM Q7 i?2b2 i2`Kb- i?Bb TT`Q+? KvMQM2i?2H2bb T`2b2Mi  T`Q?B#BiBp2 KQmMi Q7 +QKTmiiBQM Qp2`?2/ +QKT`2/iQ i?2 Q`B;BMH T`Q#H2KX AM i?2 M2ti bm#b2+iBQM A rBHH /Bb+mbb M Hi2`MiBp2b+?2K2 i?i 7Q`;Q2b mbBM; i?2 C+Q#BM M/ rBHH BMbi2/ HBM2`Bx2 i?2 T`Q#H2KBM bQK2 Qi?2` rvXhQ bmKK`Bx2 i?2 KQ/B}2/ :mbb@L2riQM TT`Q+? iQ bQHpBM; T`Q#@H2K 9X3- r2 2bb2MiBHHv M22/ iQ KQ/B7v bi2Tb k M/ j Q7 i?2 Q`B;BMH H;Q@`Bi?K #Qp2X AM bi2T k Q7 2+? Bi2`iBQM r2 2tTHB+BiHv +QKTmi2 i?2 mT/i2bgk+1 = gk + ∆gk+1 rBi? ∆gk+1 ;Bp2M #v bQHpBM; T`Q#H2K 9XRyX AM bi2T jr2 i?2M T2`7Q`K i?2 2t+i HBM2@b2`+? QM i?2 mT/i2 ∆gk+1 Bib2H7 ;BMbii?2 +m``2Mi `2bB/mHX Jm+? HBF2 BM i?2 Q`B;BMH H;Q`Bi?K- i?Bb KBiB;i2b i?2KTHBim/2 HQbb 7`QK i?2 ℓ1@MQ`K `2;mH`BxiBQM r?BH2 F22TBM; Bib bT`bB7vBM;2z2+iX3e*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;4.2.2 Relinearizationbysubstitutionb b?QrM BM 2tT`2bbBQM 9XRR- ?pBM; iQ +QKTmi2 i?2 +iBQM Q7 i?2 C+Q#BM Q7i?2 QT2`iQ` M˜ Kv #2 mM++2Ti#Hv +QbiHv- +QMbB/2`BM; i?i 2+? rp2}2H/KmHiBTHB+iBQM BM i?2b2 2tT`2bbBQMb Bb +QKTmiiBQMHHv 2[mBpH2Mi iQ QM2 7mHHa_J1 KmHiBTH2 T`2/B+iBQM bi2TX aBM+2 i?2 MQM@HBM2`Biv QMHv +QK2b 7`QK2bb2MiBHHv ?pBM; [m/`iB+ M/ ?B;?2` TQr2`b Q7G-  TQbbB#H2 Hi2`MiBp2 iQpQB/ /2HBM; rBi? i?2 C+Q#BM Bb iQ bBKTHv bm#biBimi2 i?2 T`2pBQmb Bi2`iBQMgk b  +QMbiMi BM i?2 (GRKc)n T`i Q7 2[miBQM 9XeX h?Bb `2im`Mb mb iQi?2 T`/B;K Q7 BMp2`iBM;  HBM2` QT2`iQ` BM G i bi2T k- r?B+? Bb r`Bii2MBM /i@QT2`iQ` MQiiBQM bM˜kω(G) = K[(GQk +GRP′)+ GkRKc(GQk +GRP′)+ GkRKc(GkRKc(GQk +GRP′))+ ...].U9XRkVaBKBH` iQ :mbb@L2irQM- i?2 TT`Q+? HbQ `2HBM2`Bx2b i?2 T`Q#H2K ibi2T k- 2t+2Ti i?i r2 /Q MQi ?p2 iQ }M/ 2+? mT/i2 2tTHB+BiHvX AMbi2/- r2bBKTHv `2THB+i2 bi2T k Q7 i?2 Q`B;BMH _1SaA H;Q`Bi?K UbQHpBM; i?2 GbbQT`Q#H2K BM 2[miBQM 9XN bi`iBM; 7`QK i?2 T`2pBQmb bQHmiBQM gkV mbBM; i?2TT`QtBKi2 #mi HBM2` KQ/2HHBM; QT2`iQ` Q7 2tT`2bbBQM 9XRkX AM 2z2+i- r2MQr bQHp2 i?2 7QHHQrBM; BM bi2T k,ming∥p− M˜k(g)∥2 bm#D2+i iQ ∥g∥1 < τk. U9XRjVh?2Q`2iB+HHv bT2FBM;- r2 ?p2 iQ ;Bp2 mT bQK2 MB+2 T`QT2`iB2b Q7 :mbb@L2riQM rBi? i?Bb K2i?Q/- bm+? b TQbbB#H2 [m/`iB+ +QMp2`;2M+2 M/ ;m`@Mi22b Q7 `2+?BM;  biiBQM`v TQBMiX "mi BM T`+iB+2 A }M/ i?Bb K2i?Q/-r?B+? A rBHH +HH dz`2HBM2`BxiBQMǴ 7Q` i?2 Tm`TQb2 Q7 i?Bb +?Ti2`- +QMp2`;2bbBKBH`Hv iQ i?2 :mbb@L2riQM TT`Q+? r?BH2 HbQ ?pBM; Km+? bKHH2`T2`@Bi2`iBQM +QKTmi2 +QbibX AM 7+i- i?2 +QKTmiiBQMH Qp2`?2/ ?2`2 BbK`;BMH +QKT`2/ iQ i?2 Q`B;BMH H;Q`Bi?K- b A rBHH 2tTHBM BM i?2 .Bb+mb@bBQM b2+iBQM Q7 i?Bb +?Ti2`X >Qr2p2`- iQ +?B2p2 i?Bb `2bmHi A }M/ i?i Bi BbM2+2bb`v iQ KMmHHv T`2+QM/BiBQM i?2 }`bi 72r ;`/B2Mib- r?2`2 r2 ?p2x2`Q +QMi`B#miBQM +QKBM; 7`QK i?2 KBbbBM; T`BK`v T`i Q7 p′′X3d*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;ImprovingconvergencebygradienttraceweightingA7 r2 2tTHB+BiHv r`Bi2 Qmi  UM2;iBp2V ;`/B2Mi bi2T Q7 i?2 GbbQ T`Q#H2KBM bi2T k Ur?2`2 i?2 `2bB/m2 Bb i ekV BM i?2 /i@Ki`Bt MQiiBQM- mbBM;i?2 TT`QtBKi2 QT2`iQ` BM 2tT`2bbBQM 9XRk mT iQ i?2 n=1 i2`K- r2 Q#iBM-bbmKBM; R = −I M/ mbBM; H 7Q` >2`KBiBM /DQBMi-(KEk)QHk − (KEk)P′H −KcGHk (KEk)QHk +KcGHk (KEk)P′H ._2+HH i?i Ek Bb i?2 /i@Ki`Bt `2T`2b2MiiBQM Q7 i?2 +m``2Mi `2bB/m2 2``Q`ekX 6`QK i?Bb r2 +M b22 i?i g Bb MQi #2BM; mT/i2/ 2[mHHv BMbB/2 i?2+[mBbBiBQM KbF +QKT`2/ iQ i?2 QmibB/2 Q7 BiX AM 7+i- i?2 }`bi i2`K BM i?Bbr?QH2 2tT`2bbBQM Bb +imHHv x2`Q BMbB/2 i?2 +[mBbBiBQM KbF UBX2X- KcK = 0bQ Kc(KEk)QHk = 0V- /m2 iQ i?2 `2bB/m2 #2BM; x2`Q BMbB/2 i?2 KbF- M/i?2 7+i i?i i?2 bQm`+2 rp2H2i TTHB2b  Tm`2Hv iBK2 /QKBM +QMpQHmiBQMr?B+? /Q2b MQi bTiBHHv dzbT`2/Ǵ i?2 rp2}2H/X aBM+2 i?Bb i2`K +QMiBMb HHi?2 +QMi`B#miBQM 7`QK T`BK`v rp2}2H/b- r?B+? i2M/b iQ #2 i?2 bi`QM;2biT`i Q7 i?2 rp2}2H/- i?2`2 Bb BKK2/Bi2Hv  H`;2 HQbb Q7 KTHBim/2 7Q` i?2mT/i2b BMbB/2 i?2 KbFX h?2 DQ# Q7 }HHBM; i?Bb pQB/ i?2M Mim`HHv 7HHb iQi?2 bm#b2[m2Mi i2`Kb- r?B+? HH BMpQHp2 bQK2 bQ`i Q7 rp2}2H/ +QMpQHmiBQMM/ +M TQi2MiBHHv ;2M2`i2 2p2Mib BMbB/2 i?2 KbFX >Qr2p2`- BM i?2 }`bi 72r;`/B2Mib r?2`2 Gk Bb bKHH BM KTHBim/2 U/m2 iQ i?2 ℓ1@MQ`K +QMbi`BMibV-i?2 i?B`/ i2`K M/ #Qp2 `2 HbQ mM?2HT7mHHv +HQb2 iQ x2`QX h?Bb +`2i2b +QMp2`;2M+2 dzH;Ǵ r?2`2 i`+2b Q7 g QmibB/2 Q7 i?2 KbF Kmbi H`2/v #2r2HH `2+QMbi`m+i2/ BM Q`/2` 7Q` i?2 BMbB/2 Q7 i?2 KbF iQ #2;BM mT/iBM;-r?B+? Bb T`Q#H2KiB+ #2+mb2 i?2 QmibB/2 i`+2b `2 bmb+2TiB#H2 iQ Qp2`}i QMKmHiBTH2b i?i b?QmH/ mHiBKi2Hv #2 2tTHBM2/ #v +QMi`B#miBQMb +QKBM; 7`QKi?2 BMbB/2 Q7 i?2 KbFXq2 +M KBiB;i2 i?2 BM?2`2Mi BK#HM+2 QM i?2 2`Hv ;`/B2Mib i?`Qm;? bBKTH2 r2B;?iBM; TTHB2/ iQ HH i`+2b BMbB/2 i?2 +[mBbBiBQM ;TX h?Bb+M #2 b22M b TTHvBM;  /B;QMH `B;?i T`2+QM/BiBQM2` iQ i?2 HBM2`Bx2/KQ/2HHBM; QT2`iQ`X 6Q` i?2 :mbb@L2riQM TT`Q+?- i?Bb FBM/ Q7 b+HBM;Bb miQKiB+HHv ?M/H2/ #v i?2 BKTHB+Bi >2bbBM- #mi 7Q` i?2 `2HBM2`BxiBQMK2i?Q/ r2 Kmbi 7Q`K M/ TTHv i?Bb T`2+QM/BiBQM2` KMmHHvX M p2`;2/-i`+2@BM/2T2M/2Mi p2`bBQM +M #2 Q#iBM2/ #v }M/BM; M 2t+i HBM2@b2`+? 7Q` b+HBM; α Q7 Kcgk+1 r?B+? KtBKHHv `2/m+2b i?2 `2bB/mH H27i Qp2` 7`QKi?2 +QMi`B#miBQMb Q7 Kgk+1 mM/2` i?2 +iBQM Q7 i?2 M2ti 7Q`r`/ QT2`iQ`M˜k+1 UMQi i?i Bi Bb HBM2`Bx2/ `QmM/ i?2 M2rHv Q#iBM2/ gk+1VX AM Qi?2`33*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;rQ`/b- r2 b22F i?2 b+H` α- r?B+? bQHp2b i?2 7QHHQrBM; T`Q#H2Kminα∥(p′ −KM˜k+1(Kgk+1))− αKM˜k+1(Kcgk+1)∥22.h?2 +QKTmiiBQM Q7 i?Bb b+HBM; +M #2 2z2+iBp2Hv 7`22 BM i?2 T`Q+2bb Q7+QKTmiBM; i?2 `2bB/mH Q7 Bi2`iBQM k+1- #v T`Q/m+BM; i?2 M2r T`2/B+i2/rp2}2H/ /m2 iQ i?2 irQ T`ib Q7 gk+1 b2T`i2Hv,eQmibB/2 = p′ −KM˜k+1(Kgk+1)p˜BMbB/2 = KM˜k+1(Kcgk+1)αk+1 = p˜TBMbB/2eQmibB/2/∥p˜BMbB/2∥22. U9XR9VLQi2 i?i HH i?2 +QKTmiiBQMH +Qbi 7Q` i?Bb b+HBM; Bb BM i?2 Hbi HBM2- r?B+?QMHv BMpQHp2b i?2 p2+iQ` BMM2` T`Q/m+ib #2ir22M p˜BMbB/2 M/ eQmibB/2- b r2HHb ℓ2 MQ`KbX HH i?2 i2`Kb M22/2/ iQ 7Q`K i?2b2 irQ rp2}2H/b `2 H`2/v+QKTmi2/ QM+2 r2 ?p2 i?2 }MH `2bB/m2 7Q` i?Bb Bi2`iBQMX b M bB/2-i?Bb b+HBM; b?QmH/ #2 +H+mHi2/ 7i2` i?2 HBM2@b2`+? Qp2` HH Q7 g /2b+`B#2/BM bi2T j- iQ KBMBKBx2 i?2 BKT+i Q7 i?2 ++m`+v Q7 i?2 +m``2Mi rp2H2i2biBKi2 UbBKTHv B;MQ`BM; i?2 T`BK`v GQ i2`K r?2M 7Q`KBM; p˜BMbB/2 M/eQmibB/2 rBHH ?p2  bBKBH` 2z2+iVX7i2` Q#iBMBM; α- r2 +M mb2 Bi b  `B;?i@T`2+QM/BiBQM2` αKc BM bi2T kQ7 i?2 M2ti Bi2`iBQM- r?2`2 r2 MQr bQHp2ming˜∥p− M˜k(αKcg˜)∥2 bm#D2+i iQ ∥g˜∥1 < τk. U9XR8Vh?Bb rBHH 2z2+iBp2Hv b+H2 HH ;`/B2Mi i`+2b BMbB/2 i?2 bi2M+BH #v αX 7i2` i?2BMp2`bBQM Bb +QKTH2i2- r2 Q#iBM i?2 M2r 2biBKi2 Q7 i?2 :`22MǶb 7mM+iBQM #vgk+1 = αKcg˜- r?B+? `2KQp2b i?2 BKT`BMi Q7 i?2 `B;?i T`2+QM/BiBQM2`X bi?2 Bi2`iBQM +QmMi BM+`2b2b M/ g #2+QK2b KQ`2 ++m`i2- r2 b?QmH/ b22 αi`2M/ iQr`/b RX6B;m`2 9X8 /2KQMbi`i2b i?2 2z2+iBp2M2bb Q7 i?Bb K2i?Q/X SM2H 9X8b?Qrb i?2 }`bi ;`/B2Mi mT/i2 7Q` g UbbmKBM; r2 mb2 i?2 i`m2 rp2H2i qVr2 rQmH/ ?p2 Q#iBM2/ 7`QK  7mHHv bKTH2/ /i Ub?QrM BM 6B;m`2 kXdVXSM2H 9X8# b?Qrb i?2 bK2 ;`/B2Mi bi2T i?i r2 rQmH/ Q#iBM mbBM; /irBi? KBbbBM; M2`@Qzb2ib U};m`2 9XRVX 1pB/2MiHv- i?Bb mT/i2 +QMiBMbKQbiHv +Q``2+i FBM2KiB+ BM7Q`KiBQM #Qmi i?2 rp2}2H/- #mi Bb Km+?r2F2` BM KTHBim/2 7Q` i`+2b BMbB/2 i?2 M2`@Qzb2i ;TX 7i2` TTHvBM;i?2 T`2+QM/BiBQM2` αKc- rBi? α +QKTmi2/ mbBM; i?2 BM++m`i2 ;`/B2Mi3N*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;BM SM2H 9X8# b i?2 TT`QtBKi2 bQHmiBQM- r2 b22 BM SM2H 9X8+ i?i i?BbKTHBim/2 BK#HM+2 Bb 2z2+iBp2Hv KBiB;i2/XOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4UVOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4U#VOffset (km)Time (s)−1 −0.5 0 0.5 100.20.40.60.811.21.4U+V6B;m`2 9X8 1`Hv ;`/B2Mi mT/i2b 7Q` g M22/ iQ #2 T`2+QM/BiBQM2/- Qi?2`@rBb2 +QMp2`;2M+2 rBHH #2 bHQr BMbB/2 i?2 +[mBbBiBQM KbF /m2 iQ KTHBim/2BK#HM+2bX UV i?2 2t+i }`bi ;`/B2Mi 7Q` g 7`QK 7mHHv bKTH2/ /i- ;Bp2M2t+i qX U#V i?2 }`bi ;`/B2Mi 7`QK /i rBi? KBbbBM; M2`@Qzb2ib Ub?QrM BM6B;m`2 9XRV mbBM; i?2 `2HBM2`Bx2/ 7Q`r`/ KQ/2H M˜ωX AMbB/2 i?2 M2`@Qzb2i;T- i?2 2p2Mib `2 +Q``2+i #mi ;`2iHv /BKBMBb?2/ BM KTHBim/2X U+V i?2 ;`@/B2Mi b?QrM BM U#V 7i2` TTHvBM; i?2 T`2+QM/BiBQMBM; b+HBM; /2b+`B#2/X h?2p2`;2 i`+2@BM/2T2M/2Mi b+HBM; Bb 2z2+iBp2 BM #HM+BM; i?2 KTHBim/2 Q7i?2 mT/i2b #2ir22M i?2 BMbB/2 M/ i?2 QmibB/2 Q7 i?2 M2`@Qzb2i ;TXhQ bmKK`Bx2- BM i?2 dz`2HBM2`BxiBQMǴ TT`Q+? r2 2bb2MiBHHv M22/ iQ`2TH+2 i?2 bi2Tb k M/ j Q7 i?2 Q`B;BMH H;Q`Bi?K rBi? i?2 7QHHQrBM; KQ/B}2/bi2Tb,kX aQHp2 i?2 KQ/B}2/ GbbQ T`Q#H2K U9XR8V BMbi2/ Q7 i?2 Q`B;BMHQM2 U9XNVX h?Bb `2TH+2b i?2 rp2}2H/ KQ/2HHBM; 7mM+iBQM rBi? i?2TT`QtBKi2Hv HBM2`Bx2/ p2`bBQM Q7 M˜ - /2}M2/ BM 2tT`2bbBQM 9XRk- M///b  `B;?i T`2+QM/BiBQM2` αkKcX PM+2 bQHmiBQM Bb Q#iBM2/- TTHvαkKc iQ BMp2`i i?2 T`2+QM/BiBQM2` M/ ;2i gk+1XNy*?Ti2` 9X JBiB;iBM; +[mBbBiBQM ;Tb mbBM; b+ii2`BM;jX *QKTmi2 i?2 b+HBM; 7Q` HH Q7 gk+1 b BM i?2 Q`B;BMHX 7i2` TTHv@BM; i?2 b+HBM;- +QKTmi2 i?2 M2ti T`2+QM/BiBQM2` b+HBM; αk+1 mbBM;2tT`2bbBQM 9XR9X4.3 Numerical examplesAM i?Bb b2+iBQM A /2KQMbi`i2 i?2 2{++v Q7 Kv T`QTQb2/ K2i?Q/b- 7Q+mbBM;QM i?2 KBbbBM; M2`@Qzb2i T`Q#H2KX A rBHH +QKT`2 i?2 `2bmHib ;Bp2M #v#Qi? i?2 :mbb@L2riQM M/ i?2 `2HBM2`BxiBQM TT`Q+?- mbBM; mT iQ i?2n = 2 i2`K U2tT`2bbBQM 9X8V BM i?2 m;K2Mi2/ MQMHBM2` KQ/2HHBM; QT2`iQ`M˜(g,q;p′)X A HbQ +QKT`2 Kv `2bmHib iQ i?2 2tTHB+Bi /i `2+QMbi`m+iBQMTT`Q+? 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Bb 7+BHBii2/ #v i?2 `2/m+iBQM Q7 bT2+i`H +QMi2MiX5.2.1 Expectedcomputationalsavingsduetosubsampling_Qm;?Hv bT2FBM;- _1SaA +QMp2`;2b i?`Qm;? +`27mHHv@+QMi`QHH2/ T`QD2+i2/;`/B2Mi Bi2`iBQMb- M/ #v 7` i?2 KQbi iBK2@+QMbmKBM; T`i Q7 2+? Bi2`@iBQM Bb i?2 +QKTmiiBQM Q7 M(g,q;p) M/ Bib /DQBMi QT2`iBQM iQ 7Q`K i?2;`/B2Mi mT/i2b QM gX "Qi? Q7 i?2b2 BMpQHp2  +QKTH2i2 rp2}2H/ +QMpQHm@iBQM- KQ/2HH2/ BM i?2 /Bb+`2iBx2/ b2iiBM; b Ki`Bt KmHiBTHB+iBQMb #2ir22MKQMQ@7`2[m2M+v /i@bHB+2b Q7 i?2 irQ rp2}2H/bXbbmKBM; i?i i?2 MmK#2` Q7 bKTH2b BM i?2 bQm`+2 M/ `2+2Bp2` +Q@Q`/BMi2b `2 i?2 bK2 bQ n = n`+p = nb`+- i?2 /i Ki`B+2b `2 Q7 bBx2n× n- bQ 2+? ;`/B2Mi mT/i2 Bi2`iBQM BMpQHp2b irQ QT2`iBQMb i?i `2 Q7+QKTmiiBQM +QKTH2tBiv*Qbi(n) = O(2ntn2 log nt) +O(nfn3),r?2`2 nf Bb i?2 MmK#2` Q7 /Bb+`2i2 7`2[m2M+B2b- nt Bb i?2 MmK#2` Q7 /Bb+`2i2iBK2 bKTH2b- iQ +QKTmi2 i?2 KmHiBTH2 rp2}2H/ 7Q`  k. Ub2BbKB+ HBM2V+[mBbBiBQM ;2QK2i`vX h?2 }`bi i2`K Bb i?2 +Qbi Q7 +QKTmiBM; i?2 /Bb+`2i26Qm`B2` i`Mb7Q`Kb mbBM; 66h- M/ i?2 b2+QM/ i2`K Bb i?2 +Qbi Q7 i?2 Ki`BtKmHiBTHB+iBQMbX  bBKTH2 `2/m+iBQM Q7 ;`B/ bKTH2b #v ?H7 U/Qm#HBM; i?2bKTH2 /BbiM+2V- HQM; rBi?  bBKBH` ?HpBM; Q7 i?2 ?B;?2bi 7`2[m2M+v 7`QKHQr@TbbBM;- `2bmHib BM i?2 +QKTmiiBQMH +QKTH2tBiv*Qbi(12n,12nf)=14O(2ntn2 log nt) + 116O(nfn3),r?BH2  7m`i?2` ?HpBM; UMQr QM2@[m`i2` Q7 i?2 Q`B;BMH bKTHBM;V `2bmHib BM*Qbi(14n,14nf)=116O(2ntn2 log nt) + 1128O(nfn3).1pB/2MiHv- i?2 n2 M/ n3 i2`Kb BM i?2 +QKTmiiBQMH +QKTH2tBiv Q7 1SaA;`/B2Mib r?B+? `2 MQ`KHHv  #M2 7Q` H`;2@`2 b2BbKB+ bm`p2vb MQr ?2HTbmb iQ /`biB+HHv `2/m+2 i?2 +QKTmiiBQMH +Qbi 7Q` bm#bKTH2/ /iX "vDmbi ?HpBM; i?2 bKTHBM;- 2+? ;`/B2Mi mT/i2 Bb TQi2MiBHHv +QKTmi2/ MRy3*?Ti2` 8X JmHiBH2p2H ++2H2`iBQM bi`i2;v 7Q` _Q#mbi 1SaAQ`/2` Q7 K;MBim/2 7bi2` i?M i?2 7mHH /i- M/ i QM2@[m`i2` bKTHBM; i?2bT22/mT Bb MQr TQi2MiBHHv irQ Q`/2`b Q7 K;MBim/2X AM 7+i- M 2MiB`2 _1SaAT`Q#H2K i i?2 QM2@[m`i2` bKTHBM; +M }MBb? #27Q`2 i?2 }`bi ;`/B2Mi bi2TBb 2p2M }MBb?2/ i i?2 7mHH b+H2X6Q`  r2HH@bKTH2/ j. /ib2i UTT`QtBKi2Hv i?2 bK2 bKTHBM; BM #Qi?bTiBH /B`2+iBQMb- bQ i?i n = nx`+p = ny`+p = nxb`+ = nyb`+V- i?Bb 2z2+iBb 2p2M KQ`2 /`biB+- bBM+2 i?2 n2 M/ n3 i2`Kb 7Q` i?2 k. bm`p2v bBimiBQMMQr #2+QK2b n4 M/ n6 i2`KbX q2 2M/ mT rBi? i?2 7QHHQrBM; +QKTH2tBiv 7Q`j. bm`p2vb,*Qbi(n) = O(2ntn4 log nt) +O(nfn6)*Qbi(12n,12nf)=116O(2ntn4 log nt) + 1128O(nfn6)*Qbi(14n,14nf)=1256O(2ntn4 log nt) + 18192O(nfn6)6B;m`2b 8X9 M/ 8X8 /2KQMbi`i2b i?Bb `2/m+iBQM BM +QKTmiiBQM b r2?Hp2 M/ [m`i2` i?2 Q`B;BMH bTiBH bKTHBM;X h?2 +Q`b2@;`B/ bQHmiBQMbr2`2 Km+? 7bi2` iQ +QKTmi2- rBi? i?2 Q`B;BMH T`Q#H2K iFBM; 9y KBMmi2b-kƓ /2+BKiBQM iFBM; e KBMmi2b- M/ 9Ɠ /2+BKiBQM H2bb i?M k KBMmi2bX ArBHH mb2 i?2b2 2tKTH2b iQ BHHmbi`i2 i?2 ;2M2`H KmHiBH2p2H bi`i2;v- 7i2` }`bi;BpBM;  ;2M2`H /2b+`BTiBQM M/ QmiHBM2 Q7 i?2 K2i?Q/ BM i?2 M2ti b2+iBQMX5.3 A multilevel strategy for REPSIq2 Q#b2`p2/ BM i?2 T`2pBQmb b2+iBQM i?i +Q`b2@b+H2 bQHmiBQMb Q7 _1SaA+HQb2Hv `2b2K#H2 /2+BKi2/ p2`bBQMb Q7 i?2 bQHmiBQM i i?2 Q`B;BMH bKTHBM;BMi2`pHX Ai Bb i?mb `2bQM#H2 iQ TQbimHi2 i?i i?2 +QKTmiiBQM #m/;2irQmH/ #2 KQ`2 2{+B2MiHv bT2Mi QM mT/i2b i i?2 +Q`b2` H2p2H }`bi mMiBHHH +Q`b2@b+H2 BM7Q`KiBQM Bb 2t?mbi2/- bbmKBM; i?i r2 HbQ ?p2 M2z2+iBp2 KTTBM; UBX2X- BMi2`TQHiBQMV 7`QK +Q`b2` ;`B/b iQ }M2` ;`B/bX6Q` i?2 bF2 Q7 bBKTHB+Biv M/ #`2pBiv- r2 }t  /2+BKiBQM `iBQ #2ir22Mbm#b2[m2Mi ;`B/ bBx2b- bm+? i?i r2 +M +?QQb2 Qm` BMi2`TQHiBQM K2i?Q/brBi?  FMQrM +`Bi2`BQMX AM i?Bb +?Ti2`- r2 bBKTHv +?QQb2  kƓ /2+BKiBQMM/ BMi2`TQHiBQM `iBQX hQ/v- i?Bb /2;`22 Q7 BMi2`TQHiBQM Bb `QmiBM2Hv T2`@7Q`K2/ BM a_J1 KmHiBTH2 `2KQpH T`Q+2/m`2b QM 2tBbiBM; bm`p2vbX hvTB+HHv-}2H/ /i Ui H2bi 7Q` K`BM2V rBHH ?p2  b?Qi bKTHBM; i?i Bb QM2@?H7 i?iRyN*?Ti2` 8X JmHiBH2p2H ++2H2`iBQM bi`i2;v 7Q` _Q#mbi 1SaAPosition (km)Time (s)0 0.5 1 1.5 200.20.40.60.811.21.4UV P`B;BMH bKTHBM;Position (km)Time (s)0 1 200.20.40.60.811.21.4U#V kƓPosition (km)Time (s)0 1 200.20.40.60.811.21.4U+V 9Ɠ6B;m`2 8X9 avMi?2iB+ /i b?Qi ;i?2`b i UV i?2 Q`B;BMH bTiBH bKTHBM;M/ i U#-+V irQ H2p2Hb Q7 bTiBH /2+BKiBQM- rBi? TT`QT`Bi2 iBK2@/QKBMHQr@Tbb }Hi2`b Ui jy >x M/ R8 >x `2bT2+iBp2HvV iQ KBiB;i2 `iB7+ib bT@iBH HBbBM; r?2M +QKTmiBM; i?2 bm`7+2@`2Hi2/ KmHiBTH2 rp2}2H/X h?2bKTHBM; Q7 i?2 iBK2 tBb `2KBMb mMiQm+?2/XQ7 `2+2Bp2` bKTHBM;- bQ i?Bb H2p2H Q7 BMi2`TQHiBQM Bb +QMbB/2`2/ `QmiBM2 M/#2 +?B2p2/ `2bQM#Hv r2HH #v- 2X;X- LJP@+Q``2+i2/ i`+2 p2`;BM;X _2@+m`bBQM Q7 i?Bb TT`Q+? 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Bib BMTmi /i rb HQr@Tbb }Hi2`2/ i R8>x- i?2bQHmiBQM b?QrM BM U+V `2iBMb  rB/2@#M/- dz/2+QMpQHp2/Ǵ TT2`M+2 rBi?;QQ/ `2bQHmiBQM Q7 i?2 irQ b2T`i2 2p2Mib i i4yXNbX JQ`2 BKTQ`iMiHv- i?2+Q`b2@;`B/ bQHmiBQMb r2`2 Km+? 7bi2` iQ +QKTmi2- rBi? i?2 Q`B;BMH T`Q#H2KiFBM; 9y KBMmi2b- U#V iFBM; e KBMmi2b- M/ U+V H2bb i?M k KBMmi2bXH;Q`Bi?K 8XR JmHiBH2p2H ++2H2`iBQM bi`i2;v 7Q` _1SaARX AMTmi, rp2}2H/ /i p +QMiBMBM; bm`7+2 KmHiBTH2bkX *?QQb2 KtBKmK /i /2+BKiBQM 7+iQ` 2S - b2i b+H2 p`B#H2 s=SjX a2i BMBiBH bQHmiBQM gs=S iQ x2`Q p2+iQ`9X _2T2i8X ps ← bm#bKTH2 Q`B;BMH /i BM HH bTiBH +QQ`/BMi2b #v 7+iQ` Q7 2SeX GQr@Tbb }Hi2` BM i?2 iBK2 /QKBM QM ps iQ `2KQp2 bTiBH HBbBM;dX aQHp2 i?2 _1SaA T`Q#H2K QM bm#bKTH2/ /i ps- BMBiBHBxBM; 7`QK gs3X gs−1 ← BMi2`TQHi2 bQHmiBQM gs BM HH bTiBH +QQ`/BMi2b #v 7+iQ` Q7 kNX s ← s− 1 U;Q iQ }M2` b+H2VRyX mMiBH b+H2 p`B#H2 s=0 U`2+?2/ Q`B;BMH bTiBH bKTHBM;VRRX aQHp2 i?2 _1SaA T`Q#H2K QM Q`B;BMH /i p- BMBiBHBxBM; 7`QK g0RkX PmiTmi, TT`QtBKi2 bQHmiBQM g˜ iQ i?2 _1SaA T`Q#H2K U2[miBQM kX9VRRR*?Ti2` 8X JmHiBH2p2H ++2H2`iBQM bi`i2;v 7Q` _Q#mbi 1SaA0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.500.51Time (s)Zero−offset trace, 1140m  IR ReferenceIR 1:2 trace, lowpass 30HzIR 1:4 trace, lowpass 15Hz6B;m`2 8Xe w2`Q@Qzb2i i`+2 7Q` i?2 +QKTmi2/ _1SaA bQHmiBQMb b?QrM BM6B;m`2 8X8X b +M #2 b22M- KQbi Q7 i?2 KDQ` 2p2Mib +QBM+B/2- Hi?Qm;?MQiB+2#H2 /2;`/iBQM +M #2 b22M BM i?2 R8>x bQHmiBQM- bm;;2biBM; i?i7m`i?2` bm#bKTHBM; KB;?i T`Q/m+2 `2bmHib rBi? +QKTH2i2Hv mM`2+Q;MBx#H22p2MibX5.4 Numerical exampleAM i?Bb b2+iBQM A rBHH BHHmbi`i2 i?2 KmHiBH2p2H bi`i2;v mbBM; i?2 7KBHB`bvMi?2iB+ /ib2i b?QrM BM i?2 T`2pBQmb b2+iBQMbX A HbQ mb2 i?2 /2b+`BTiBQMbBM i?Bb b2+iBQM iQ 2tTM/ QM bQK2 T`+iB+H /2iBHb Q7 i?2 KmHiBH2p2H bi`i2;v-b r2HH b ;Bp2 iM;B#H2 };m`2b 7Q` +QKTmiiBQMH iBK2 bpBM;bX A rBHH HbQmb2  LQ`i? a2 /ib2i bBKBH` iQ i?2 QM2 mb2/ BM i?2 Hbi b2+iBQM U#mi rBi?T`2@BMi2`TQHi2/ M2`@Qzb2ibV iQ /2KQMbi`i2 i?2 pB#BHBiv Q7 i?Bb K2i?Q/ BMT`+iB+H TTHB+iBQM- 2p2M i?Qm;? i?2 BMi2`TQHiBQM M/ /2+BKiBQM iQQHbmb2/ ?2`2 `2 `2HiBp2Hv bBKTH2XRRk*?Ti2` 8X JmHiBH2p2H ++2H2`iBQM bi`i2;v 7Q` _Q#mbi 1SaA5.4.1 SyntheticdataexamplelbBM; i?2 7mHHv@bKTH2/- }t2/@bT`2/ k. bvMi?2iB+ b2BbKB+ HBM2 /i b?QrMBM 6B;m`2 8X9- A BHHmbi`i2 i?2 T2`7Q`KM+2 BKT`Qp2K2Mib TQbbB#H2 rBi? 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QM2@bB/2/ >MMBM; rBM/Qr i jy>xX aBKBH`Hv- 6B;m`2 8X8+ 7QHHQrb7`QK i?2 /i b?QrM BM 6B;m`2 8X9+ rBi? eyK bTiBH bKTHBM; M/ HQr@Tbbi R8 >xX h?2 i?`22 /Bz2`2Mi bQHmiBQMb BM 6B;m`2 8X8 HH iQQF `Qm;?Hv d8Bi2`iBQMb Q7 i?2 _1SaA H;Q`Bi?KX .m2 iQ i?2 +m#B+ b+HBM; Q7 +QKTmi2QT2`iBQMb rBi? i?2 MmK#2` Q7 ;`B/ TQBMib- Q#iBMBM; i?2 +Q`b2bi@b+H2 bQ@HmiBQM BM 6B;m`2 8X8+ Bb `Qm;?Hv 2[mH BM `mMiBK2 iQ Dmbi R Bi2`iBQM Q7 i?2Q`B;BMH mM/2+BKi2/ T`Q#H2KXq2 MQr 7QHHQr i?2 KmHiBH2p2H bi`i2;v QmiHBM2/ BM H;Q`Bi?K 8XR- bi`i@BM; rBi? i?2 +Q`b2bi /2+BKiBQM 7+iQ` 9Ɠ U+Q``2bTQM/BM; iQ s = 2VX b?BMi2/ BM 6B;m`2 8Xe- /2+BKiBQM 7+iQ`b +Q`b2` i?M i?Bb T`Q/m+2/ :`22MǶb7mM+iBQMb i?i `2 iQQ 2``QM2Qmb iQ +i b  ;QQ/ bi`iBM; KQ/2HX h?2 1SaAT`Q#H2K Bb bQHp2/ i i?Bb b+H2 Ub?QrM #+F BM 6B;m`2 8X8+V- mbBM; d8 Bi2`@iBQMb b /2b+`B#2/ #Qp2X h?Bb iF2b `Qm;?Hv k KBMmi2b QM  +QMi2KTQ``vrQ`FbiiBQMX h?2 :`22MǶb 7mM+iBQM bQHmiBQM i 9Ɠ /2+BKiBQM Bb i?2M KTT2/iQ i?2 M2ti }M2` ;B`/ Q7 kƓ /2+BKiBQM U+Q``2bTQM/BM; iQ s = 1V pB  bBKTH2HBM2` BMi2`TQHiBQM 7i2` TTHvBM; LJP +Q``2+iBQM BM i?2 +QKKQM@KB/TQBMi/QKBM UmbBM;  +QMbiMi p2HQ+Biv Q7 Reyy KfbV- r?B+? 2z2+iBp2Hv /2+`2b2bi?2 KtBKmK bTiBH rp2MmK#2` BM  /2i2`KBMBbiB+ rv #v `2/m+BM; i?2+m`pim`2 Q7 ?vT2`#QHB+ rp27`QMibX 7i2` BMi2`TQHiBQM- M BMp2`b2 LJP+Q``2+iBQM Bb i?2M TTHB2/XRRj*?Ti2` 8X JmHiBH2p2H ++2H2`iBQM bi`i2;v 7Q` _Q#mbi 1SaAh?2 `2bmHi Q7 i?Bb BMi2`TQHiBQM M/ KTTBM; iQ i?2 kƓ ;`B/ Bb b?QrMBM 8XdX A i?2M `mM i?2 _1SaA H;Q`Bi?K mbBM; i?2 kƓ /2+BKi2/ M/ HQr@Tbb }Hi2`2/ /i b?QrM BM 6B;m`2 8X9#- mbBM; i?2 BMi2`TQHi2/ 9Ɠ bQHmiBQM bi?2 BMBiBH bQHmiBQM Q7 i?2 Bi2`iBp2 H;Q`Bi?KX h?Bb rQmH/ HbQ K2M bFBTTBM;i?2 Q`B;BMH bTBF2@TB+FBM; BMBiBHBxiBQM bi2T QmiHBM2/ BM a2+iBQM jXjX8X LQi2i?i i?2 bQHmiBQM 7Q` i?2 rp2H2i q i 9Ɠ M22/b iQ #2 /Bb+`/2/- bBM+2 Bi rBHH?p2 i?2 bT2+i`H BKT`BMi Q7 i?2 R8 >x HQr@Tbb }Hi2` r?B+? Bb TTHB2/ iQi?2 9Ɠ /2+BKi2/ /i- M/ Bb i?mb BMpHB/ 7Q` i?2 kƓ /2+BKi2/ /i i?iBb HQr@Tbb }Hi2`2/ i jy >xX  M2r BMBiBH rp2H2i Kmbi #2 i?2`27Q`2 #22biBKi2/ BMbi2/ 7`QK i?2 BMi2`TQHi2/ bQHmiBQMX7i2` jy Bi2`iBQMb i i?2 kƓ ;`B/- r2 Q#iBM  bQHmiBQM i?i Bb +QKT`#H2iQ i?2 KBb}i +?B2p2/ 7i2` d8 Bi2`iBQMb i kƓ ;`B/ bi`iBM; 7`QK M 2KTivBMBiBH bQHmiBQMX h?2 `2bmHiBM; :`22MǶb 7mM+iBQM Bb b?QrM BM 6B;m`2 8Xd#-r?B+? +M #2 +QKT`2/ iQ i?2 `272`2M+2 bQHmiBQM 7Q` i?Bb b+H2 b?QrM BM6B;m`2 8X8#XPosition (km)Time (s)0 1 200.20.40.60.811.21.4UV BMBiBH i kƓPosition (km)Time (s)0 1 200.20.40.60.811.21.4U#V }MH i kƓ6B;m`2 8Xd h?2 :`22MǶb 7mM+iBQM KQ/2H i i?2 kƓ /2+BKiBQM Ujy KV bK@THBM;- +QKTmi2/ b T`i Q7  KmHiBH2p2H +QMiBMmiBQM bi`i2;v 7Q` ++2H2`iBM;_1SaAX SM2H UV b?Qrb i?2 LJP@BMi2`TQHi2/ bQHmiBQM Q7 i?2 _1SaA T`Q#@H2K i 9Ɠ /2+BKiBQM Ub?QrM BM 6B;m`2 8X8+V- r?B+? 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Bb bBKBH` iQ i?2 bvMi?2iB+ +b2-M/ H`;2Hv BM~m2M+2/ #v i?2 MmK#2` Q7 Bi2`iBQMb i i?2 Q`B;BMH b+H2 BMi?2 }MH Qmi2` HQQTXRRe*?Ti2` 8X JmHiBH2p2H ++2H2`iBQM bi`i2;v 7Q` _Q#mbi 1SaAPosition (km)Time (s)0 0.5 1 1.5 200.20.40.60.811.21.4UV _272`2M+2 T`BK`vPosition (km)Time (s)0 0.5 1 1.5 200.20.40.60.811.21.4U#V JmHiBH2p2H bi`i2;v T`BK`vPosition (km)Time (s)0 0.5 1 1.5 200.20.40.60.811.21.4U+V _272`2M+2 KmHiBTH2Position (km)Time (s)0 0.5 1 1.5 200.20.40.60.811.21.4U/V JmHiBH2p2H bi`i2;v KmHiBTH26B;m`2 8XRy avMi?2iB+ 2tKTH2 +QKT`BbQM b?Qi@;i?2`b Q7 i?2 }MH /2KmH@iBTH2/ T`BK`v rp2}2H/ #2ir22M i?2 _1SaA H;Q`Bi?K `mM i i?2 Q`B;BMHbTiBH bKTHBM;- M/ i?2 T`QTQb2/ KmHiBH2p2H bi`i2;v #2;BMMBM; i  9Ɠ/2+BKiBQM Q7 i?2 /iXRRd*?Ti2` 8X JmHiBH2p2H ++2H2`iBQM bi`i2;v 7Q` _Q#mbi 1SaA00.51.01.52.0Time (s)0 1 2 3 4 5Rcv position (km)UV P`B;BMH bKTHBM;00.51.01.52.0Time (s)0Rcv position (km)U#V kƓ00.51.01.52.0Time (s)0Rcv position (km)U+V 9Ɠ6B;m`2 8XRR LQ`i? a2 }2H/ /i b?Qi ;i?2`b i UV i?2 Q`B;BMH bTiBHbKTHBM; M/ i U#-+V irQ H2p2Hb Q7 bTiBH /2+BKiBQM- rBi? 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HBKBiiBQM BKTQb2/ #vi?2 HQr@Tbb }Hi2` #2+QK2b KQ`2 /mMiBM;- bQ #2BM; #H2 iQ ;2i  KQ`2++m`i2 bQHmiBQM 7`QK  +QKTmiiBQMHHv +QbiHv H;Q`Bi?K +M #2 r2H+QK2XA }M/ i?Bb HBM2 Q7 `2b2`+? p2`v T`QKBbBM;- M/ /2}MBi2Hv THM iQ Tm`bm2 i?BbBM[mB`v BM i?2 7mim`2XRkeBibliographyHJi`- JX >X- kyRy- 1biBKiBQM Q7 am`7+2@7`22 .i #v *m`p2H2i@/QKBMJi+?2/ 6BHi2`BM; M/ aT`b2 AMp2`bBQM, Jbi2`Ƕb i?2bBb- lMBp2`bBiv Q7"`BiBb? *QHmK#BX → T;2b R3- RkeKmM/b2M- GX- kyyR- 1HBKBMiBQM Q7 7`22@bm`7+2 `2Hi2/ KmHiBTH2b rBi?QmiM22/ Q7 i?2 bQm`+2 rp2H2i, :2QT?vbB+b- ee- jkdĜj9RX → T;2b kjMbi2v- LX X- M/ SX L2rKM- RNee- S`i A, h?2 b2+iBQMHmiQ@+Q``2HQ;`K M/ S`i AA, h?2 b2+iBQMH `2i`Q@+Q``2HQ;`K,:2QT?vbB+H S`QbT2+iBM;- R9- j3NĜ9keX → T;2b R3- ky`pFBM- X uX- M/ hX pM G22mr2M- kyRk- 1biBKiBM; LmBbM+2S`K2i2`b BM AMp2`b2 S`Q#H2Kb, AMp2`b2 S`Q#H2Kb- k3- ReX → T;2b 8y"`/KM- _X >X- .X CX o2`b+?mm`- _X :X pM "Q`b2H2M- JX PX 6`BDHBMF- M/_X 6X >2;;2- kyRy- 1biBKiBQM Q7 T`BK`B2b #v bT`b2 BMp2`bBQM mbBM;/mH@b2MbQ` /i, a1: h2+?MB+H S`Q;`K 1tTM/2/ #bi`+ib- kN-j9e3X → T;2b ke"+Fmb- JX JX- RN8N- qi2` `2p2`#2`iBQM @ i?2B` Mim`2 M/ 2HBKBMiBQM,:2QT?vbB+b- k9- kjjĜkeRX → T;2b R9"iiH2- :X- RN3d-  #HQ+F bTBM +QMbi`m+iBQM Q7 QM/2H2ii2bX S`i A, G2K`Bû7mM+iBQMb, *QKKmMB+iBQMb BM Ji?2KiB+H S?vbB+b- RRy- eyRĜeR8X →T;2b 8N"2`F?Qmi- X CX- M/ uX >X SQ- RN3k- a2BbKB+ JB;`iBQM @ AK;BM; Q7+QmbiB+ 1M2`;v #v qp2 6B2H/ 1ti`TQHiBQM, CQm`MH Q7 TTHB2/J2+?MB+b- 9N- e3kX → T;2b kR- k9Rkd"B#HBQ;`T?v"2`F?Qmi- X CX- M/ .X CX o2`b+?mm`- RNNd- 1biBKiBQM Q7 KmHiBTH2b+ii2`BM; #v Bi2`iBp2 BMp2`bBQM- S`i A, h?2Q`2iB+H +QMbB/2`iBQMb,:2QT?vbB+b- ek- R83eĜR8N8X → T;2b kR"2`Mi?- >X- M/ GX aQMM2HM/- RN3j- qp2@}2H/ 2ti`TQHiBQM i2+?MB[m2b7Q` T`2bi+F ii2MmiBQM Q7 ri2` `2p2`#2`iBQMb, a1: h2+?MB+HS`Q;`K 1tTM/2/ #bi`+ib- ke9Ĝke8X → T;2b RN"2``v?BHH- CX _X- M/ uX *X EBK- RN3e- .22T@ri2` T2; H2;b M/ KmHiBTH2b,1KmHiBQM M/ bmTT`2bbBQM, :2QT?vbB+b- 8R- kRddĜkR39X → T;2b RN"2x/2F- CX *X- _X CX >i?rv- _X 1X >Qr`/- *X X qBHbQM- M/ JX SXqBM/?K- RN3d- GQ+H +QMp2`;2M+2 MHvbBb Q7  ;`QmT2/ p`B#H2p2`bBQM Q7 +QQ`/BMi2 /2b+2Mi, CQm`MH Q7 PTiBKBxiBQM h?2Q`v M/TTHB+iBQMb- 89- 9dRĜ9ddX → T;2b 9k"B2`bi2F2`- CX- kyyR- J:A*, a?2HHǶb bm`7+2 KmHiBTH2 ii2MmiBQMi2+?MB[m2, a1: h2+?MB+H S`Q;`K 1tTM/2/ #bi`+ib kyyR- aQ+B2iv Q71tTHQ`iBQM :2QT?vbB+Bbib- RjyRĜRjy9X → T;2b kk- kj"B;;b- .X aX *X- RNN3- ++2H2`i2/ Bi2`iBp2 #HBM/ /2+QMpQHmiBQM, S?.i?2bBb- lMBp2`bBiv Q7 m+FHM/- m+FHM/X → T;2b 8j"B`;BM- 1X :X- CX JX J`iBM2x- M/ JX _v/M- kyyy- LQMKQMQiQM2bT2+i`H T`QD2+i2/ ;`/B2Mi K2i?Q/b QM +QMp2t b2ib, aAJ CQm`MH QMPTiBKBxiBQM- Ry- RRNeĜRkRRX → T;2b 39- Ne*#`2HHB- *X X- RN38- JBMBKmK 2Mi`QTv /2+QMpQHmiBQM M/ bBKTHB+Biv, MQMBi2`iBp2 H;Q`Bi?K, :2QT?vbB+b- 8y- jN9X → T;2b kN*K#QBb- :X- .X *`HbQM- *X CQM2b- JX G2bM2b- qX aƺHHM2`- M/ >X h#iB-kyyN- .mH@b2MbQ` bi`2K2` /i, *HB#`iBQM- +[mBbBiBQM Z* M/ii2MmiBQM Q7 b2BbKB+ BMi2`72`2M+2b M/ Qi?2` MQBb2b, a1: h2+?MB+HS`Q;`K 1tTM/2/ #bi`+ib- R9kĜR9eX → T;2b N8*M/ĕb- 1X CX- CX _QK#2`;- M/ hX hQ- kyye- ai#H2 bB;MH `2+Qp2`v 7`QKBM+QKTH2i2 M/ BM++m`i2 K2bm`2K2Mib, *QKKmMB+iBQMb QM Sm`2 M/TTHB2/ Ji?2KiB+b- 8N- RkydX → T;2b 9d*?2M- aX aX- M/ .X GX .QMQ?Q- RNN9- "bBb Tm`bmBi, S`Q+22/BM;b Q7 k3i?bBHQK` *QM72`2M+2 QM aB;MHb- avbi2Kb M/ *QKTmi2`b- A111 *QKTmiXaQ+X S`2bb- 9RĜ99X → T;2b j8Rk3"B#HBQ;`T?v*?2M- aX aX- .X GX .QMQ?Q- M/ JX X amM/2`b- kyyR- iQKB+/2+QKTQbBiBQM #v #bBb Tm`bmBi, aAJ _2pB2r- 9j- RkNX → T;2b j8- 98*H2`#Qmi- CX 6X- M/ 6X JmB`- RNdj- _Q#mbi KQ/2HBM; rBi? 2``iB+ /i,:2QT?vbB+b- j3- 3keX → T;2b 9e*xv+xmH _m/DQ`/- wX- EX /2 oQb- M/ SX >QQ;2`#`m;;2- kyR8- _2H .i*QKT`BbQM Q7 1SaA M/ _Q#mbi 1SaA AM+Hm/BM; i?2 L2` Pzb2i_2+QMbi`m+iBQM, S`2b2Mi2/ i i?2 ddi? 1:1 *QM72`2M+2  1t?B#BiBQM-1:1X → T;2b Rk9.m#2+?B2b- AX- JX 6Q`MbB2`- M/ AX GQ`Bb- kyy3- ++2H2`i2/ T`QD2+i2/;`/B2Mi K2i?Q/ 7Q` HBM2` BMp2`b2 T`Q#H2Kb rBi? bT`bBiv +QMbi`BMib,CQm`MH Q7 6Qm`B2` MHvbBb M/ TTHB+iBQMb- R9- de9ĜdNkX → T;2b je-39- RjN- R9R.QMMQ- .X- kyRR- AKT`QpBM; KmHiBTH2 `2KQpH mbBM; H2bi@b[m`2b /BT }Hi2`bM/ BM/2T2M/2Mi +QKTQM2Mi MHvbBb, :2QT?vbB+b- de- oNRĜoRy9X →T;2b kk.QMMQ- .X- >X *?m`Bb- M/ JX LQ#H2- kyRy- *m`p2H2i@#b2/ KmHiBTH2T`2/B+iBQM, :2QT?vbB+b- d8- q"k88Ĝq"kejX → T;2b kk.QMQ?Q- .X GX- RN3R- PM KBMBKmK 2Mi`QTv /2+QMpQHmiBQM- BM TTHB2/ iBK2b2`B2b MHvbBb AA, +/2KB+ S`2bb- 8e8Ĝey3X → T;2b kNěěĜ- kyye- 6Q` KQbi H`;2 mM/2`/2i2`KBM2/ bvbi2Kb Q7 HBM2` 2[miBQMbi?2 KBMBKH GR@MQ`K bQHmiBQM Bb HbQ i?2 bT`b2bi bQHmiBQM,*QKKmMB+iBQMb QM Sm`2 M/ TTHB2/ Ji?2KiB+b- 8N- dNdĜ3kNX →T;2b 9e- 9d.QMQ?Q- .X GX- M/ uX hbB;- kyy3- 6bi aQHmiBQM Q7 GR@LQ`K JBMBKBxiBQMS`Q#H2Kb q?2M i?2 aQHmiBQM Jv "2 aT`b2, A111 h`Mb+iBQMb QMAM7Q`KiBQM h?2Q`v- 89- 9d3NĜ93RkX → T;2b 93.QbbH- *X- M/ aX JHHi- kyy8- aT`b2 bTBF2 /2+QMpQHmiBQM rBi? KBMBKmKb+H2, S`Q+X aB;MH S`Q+2bbBM; rBi? /TiBp2 aT`b2 ai`m+im`2/_2T`2b2MiiBQMb kyy8- RkjĜRkeX → T;2b 9e.`;Qb2i- "X- AX JQQ`2- M/ *X EQbiQp- kyye- h?2 BKT+i Q7 }2H/@bm`p2v+?`+i2`BbiB+b QM bm`7+2@`2Hi2/KmHiBTH2 ii2MmiBQM, :2QT?vbB+HS`QbT2+iBM;- 89- d3RĜdNRX → T;2b RykRkN"B#HBQ;`T?v.`;Qb2i- "X- .X CX o2`b+?mm`- AX JQQ`2- M/ _X "BbH2v- kyRy-  T2`bT2+iBp2QM j. bm`7+2@`2Hi2/ KmHiBTH2 2HBKBMiBQM, :2QT?vbB+b- d8- d8k98X →T;2b kk.`;Qb2i- qX >X- M/ pX C2`BÍ2pBÉ- RNN3- aQK2 `2K`Fb QM bm`7+2 KmHiBTH2ii2MmiBQM, :2QT?vbB+b- ej- ddkĜd3NX → T;2b kk.m``MB- CX X- RNNR- JmHiBTH2 ii2MmiBQM #v T`2/B+iBp2 /2+QMpQHmiBQM BMi?2 T@i /QKBM, S`2b2Mi2/ i i?2 1:1 h2+?MB+H S`Q;`K 1tTM/2/#bi`+ibX → T;2b R81bb2`- 1X- hX GBM- _X qM;- M/ 6X >2``KMM- kyR8-  GB7i2/ HRfHk*QMbi`BMi 7Q` aT`b2 "HBM/ .2+QMpQHmiBQM, S`2b2Mi2/ i i?2 ddi?1:1 *QM72`2M+2  1t?B#BiBQM- 1:1X → T;2b Rke6B;m2B`2/Q- JX X hX- _X .X LQrF- M/ aX CX q`B;?i- kyyd- :`/B2MiS`QD2+iBQM 7Q` aT`b2 _2+QMbi`m+iBQM, TTHB+iBQM iQ *QKT`2bb2/a2MbBM; M/ Pi?2` AMp2`b2 S`Q#H2Kb, A111 CQm`MH Q7 a2H2+i2/ hQTB+b BMaB;MH S`Q+2bbBM;- R- 83eĜ8NdX → T;2b 8R6H2Bb+?2`- :X- _X :Q`2M~Q- M/ "X >Q7KMM- RNNN- PM i?2 miQ+QMpQHmiBQM1[miBQM M/ hQiH o`BiBQM *QMbi`BMib, wJJ- dN- R9NĜR8NX →T;2b 386H2Bb+?2`- :X- M/ "X >Q7KMM- RNNe- PM BMp2`bBQM `i2b 7Q` i?2miQ+QMpQHmiBQM 2[miBQM, AMp2`b2 S`Q#H2Kb- Rk- 9RNĜ9j8X → T;2b 386QFF2K- CX hX- M/ SX JX oM /2M "2`;- RNNy- _2KQpH Q7 bm`7+2ɢ`2Hi2/rp2 T?2MQK2M, h?2 K`BM2 +b2, a1: h2+?MB+H S`Q;`K 1tTM/2/#bi`+ib- Re3NĜReNkX → T;2b kyěěĜ- RNNj- a2BbKB+ TTHB+iBQMb Q7 +QmbiB+ `2+BT`Q+Biv, 1Hb2pB2` a+B2M+2X→ T;2b Rk- ky6Qbi2`- .X CX- M/ *X *X JQb?2`- RNNk- amTT`2bbBQM Q7 KmHiBTH2 `2~2+iBQMbmbBM; i?2 _/QM i`Mb7Q`K, :2QT?vbB+b- 8d- j3eĜjN8X → T;2b R86`BDHBMF- JX PX- _X :X pM "Q`b2H2M- M/ qX aƺHHM2`- kyRR- h?2 7`22 bm`7+2bbmKTiBQM 7Q` K`BM2 /i@/`Bp2M /2KmHiBTH2 K2i?Q/b, :2QT?vbB+HS`QbT2+iBM;- 8N- keNĜkd3X → T;2b R3- ky- RRy6m+?b- CX- kyy8- _2+Qp2`v Q7 1t+i aT`b2 _2T`2b2MiiBQMb BM i?2 S`2b2M+2Q7 "QmM/2/ LQBb2, A111 h`Mb+iBQMb QM AM7Q`KiBQM h?2Q`v- 8R-jeyRĜjey3X → T;2b 9dRjy"B#HBQ;`T?v:mBiiQM- X- M/ .X CX o2`b+?mm`- kyy9- /TiBp2 bm#i`+iBQM Q7 KmHiBTH2bmbBM; i?2 GR@MQ`K, :2QT?vbB+H S`QbT2+iBM;- 8k- kdĜj3X → T;2b kk- kj-j9>/B/B- JX hX- JX a#B?- .X 1X CQ?MbiQM- M/ *X *H/2`ƦM@J+b- RNNN-JQ#BHǶb `2bmHib 7Q` i?2 RNNd rQ`Fb?QT QM KmHiBTH2 ii2MmiBQM, h?2G2/BM; 1/;2- R3- RyyĜRyjX → T;2b kk>H2- 1X hX- qX uBM- M/ uX w?M;- kyy3- 6Bt2/@SQBMi *QMiBMmiBQM 7Q`GR@JBMBKBxiBQM, J2i?Q/QHQ;v M/ *QMp2`;2M+2, aAJ CQm`MH QMPTiBKBxiBQM- RN- RRydĜRRjyX → T;2b je- 9d>KTbQM- .X- RN3e- AMp2`b2 o2HQ+Biv ai+FBM; 7Q` JmHiBTH2 1HBKBMiBQM,CQm`MH Q7 i?2 *M/BM aQ+B2iv Q7 1tTHQ`iBQM :2QT?vbB+Bbib- kk- 99Ĝ98X→ T;2b R8>2MM2M72Mi- :X- GX 62M2HQM- M/ 6X CX >2``KMM- kyRy- LQM2[mBbT+2/+m`p2H2i i`Mb7Q`K 7Q` b2BbKB+ /i `2+QMbi`m+iBQM,  bT`bBiv@T`QKQiBM;TT`Q+?, :2QT?vbB+b- d8- q"kyjĜq"kRyX → T;2b d9>2MM2M72Mi- :X- 1X pM /2M "2`;- JX SX 6`B2/HM/2`- M/ 6X CX >2``KMM-kyy3- L2r BMbB;?ib BMiQ QM2@MQ`K bQHp2`b 7`QK i?2 S`2iQ +m`p2,:2QT?vbB+b- dj- kjX → T;2b je- 39- RjN>2``KMM- 6X CX- kyy8- a2BbKB+ /2+QMpQHmiBQM #v iQKB+ /2+QKTQbBiBQM, T`K2i`B+ TT`Q+? rBi? bT`b2M2bb +QMbi`BMib, AMi2;`i2/*QKTmi2`@B/2/ 1M;BM22`BM;- Rk- eNĜNyX → T;2b 9eěěĜ- kyRy- _M/QKBx2/ bKTHBM; M/ bT`bBiv, :2iiBM; KQ`2 BM7Q`KiBQM7`QK 72r2` bKTH2b, :2QT?vbB+b- d8- q"RdjĜq"R3dX → T;2b 8k>2``KMM- 6X CX- lX "Q2MB;2`- M/ .X CX o2`b+?mm`- kyyd- LQM@HBM2`T`BK`v@KmHiBTH2 b2T`iBQM rBi? /B`2+iBQMH +m`p2H2i 7`K2b,:2QT?vbB+H CQm`MH AMi2`MiBQMH- Rdy- d3RĜdNNX → T;2b kk>2``KMM- 6X CX- uX X 1`HM;;- M/ hX hX uX GBM- kyyN- *QKT`2bbBp2bBKmHiM2Qmb 7mHH@rp27Q`K bBKmHiBQM, :2QT?vbB+b- d9- j8X → T;2bey- dy>2``KMM- 6X CX- M/ :X >2MM2M72Mi- kyy3- LQM@T`K2i`B+ b2BbKB+ /i`2+Qp2`v rBi? +m`p2H2i 7`K2b, :2QT?vbB+H CQm`MH AMi2`MiBQMH- Rdj-kjjĜk93X → T;2b d9RjR"B#HBQ;`T?v>2``KMM- 6X CX- .X qM;- M/ .X CX o2`b+?mm`- kyy3- /TiBp2+m`p2H2i@/QKBM T`BK`v@KmHiBTH2 b2T`iBQM, :2QT?vbB+b- dj- RdĜkRX→ T;2b kk>2``KMM- SX- hX JQD2bFv- JX J;2bM- M/ SX >m;QMM2i- kyyy-.2ɢHBb2/- ?B;?ɢ`2bQHmiBQM `/QM i`Mb7Q`Kb, a1: h2+?MB+H S`Q;`K1tTM/2/ #bi`+ib- RN8jĜRN8eX → T;2b R8CmK?- "X- M/ 6X CX >2``KMM- kyRR- .BK2MbBQMHBiv@`2/m+2/ 2biBKiBQMQ7 T`BK`B2b #v bT`b2 BMp2`bBQM, a1: h2+?MB+H S`Q;`K 1tTM/2/#bi`+ib- jy- j8kyX → T;2b dyE#B`- JX LX- M/ .X o2`b+?mm`- RNN8- _2biQ`iBQM Q7 KBbbBM; Qzb2ib #vT`#QHB+ _/QM i`Mb7Q`KR, :2QT?vbB+H S`QbT2+iBM;- 9j- j9dĜje3X →T;2b d9ETHM- aX hX- M/ EX X AMMM2M- kyy3- /TiBp2 b2T`iBQM Q77`22@bm`7+2 KmHiBTH2b i?`Qm;? BM/2T2M/2Mi +QKTQM2Mi MHvbBb,:2QT?vbB+b- dj- okNĜojeX → T;2b kkE2HKBb- SX :X- 1X 6X *?B#m`Bb- M/ aX a??`v`- RNNy- _/QM KmHiBTH22HBKBMiBQM-  T`+iB+H K2i?Q/QHQ;v 7Q` HM/ /iX → T;2b R8E2MM2ii- "X GX LX- RNdN- h?2 bmTT`2bbBQM Q7 bm`7+2 KmHiBTH2b QM b2BbKB+`2+Q`/b, :2QT?vbB+H S`QbT2+iBM;- kd- 839ĜeyyX → T;2b kyE`2BK2`- LX- X aiMiQM- M/ JX .X a++?B- kyRj- h2MbQ` +QKTH2iBQM #b2/QM Mm+H2` MQ`K KBMBKBxiBQM 7Q` 8. b2BbKB+ /i `2+QMbi`m+iBQM,:2QT?vbB+b- d3- okdjĜok39X → T;2b d9EmK`- _X- *X .X aBHp- PX FHBM- X uX `pFBM- >X JMbQm`- "X _2+?i-M/ 6X CX >2``KMM- kyR8- 1{+B2Mi Ki`Bt +QKTH2iBQM 7Q` b2BbKB+ /i`2+QMbi`m+iBQM, :2QT?vbB+bX → T;2b d9G2pBM- aX X- RN3d- .2+QMpQHmiBQM rBi? bTiBH +QMbi`BMib, S?. i?2bBb-aiM7Q`/ lMBp2`bBiv- aiM7Q`/- *X → T;2b RNGB- sX- X uX `pFBM- hX pM G22mr2M- M/ 6X CX >2``KMM- kyRk- 6bi`M/QKBx2/ 7mHH@rp27Q`K BMp2`bBQM rBi? +QKT`2bbBp2 b2MbBM;,:2QT?vbB+b- dd- RjĜRdX → T;2b 38GB- wX@tX- M/ qX@FX Gm- kyRj- /TiBp2 KmHiBTH2 bm#i`+iBQM #b2/ QM j.#HBM/ b2T`iBQM Q7 +QMpQHp2/ KBtim`2b, :2QT?vbB+b- d3- ok8RĜokeeX →T;2b kkRjk"B#HBQ;`T?vGBM- hX hX uX- M/ 6X CX >2``KMM- kyyN- lMB}2/ +QKT`2bbBp2 b2MbBM;7`K2rQ`F 7Q` bBKmHiM2Qmb +[mBbBiBQM rBi? T`BK`v 2biBKiBQM, a1:h2+?MB+H S`Q;`K 1tTM/2/ #bi`+ib- a1:- jRRjĜjRRdX → T;2b dyěěĜ- kyRR- _Q#mbi bQm`+2 bB;Mim`2 /2+QMpQHmiBQM M/ i?2 2biBKiBQM Q7T`BK`B2b #v bT`b2 BMp2`bBQM, a1: h2+?MB+H S`Q;`K 1tTM/2/#bi`+ib- 9j89X → T;2b RkeGQFb?iMQp- .X- RNNN- JmHiBTH2 bmTT`2bbBQM #v /iɢ+QMbBbi2Mi/2+QMpQHmiBQM, h?2 G2/BM; 1/;2- R3- RR8ĜRRNX → T;2b R8ěěĜ- kyyy- amTT`2bbBQM Q7 ri2`@Hv2` KmHiBTH2bě7`QK /2+QMpQHmiBQM iQrp2@2[miBQM TT`Q+?, a1: h2+?MB+H S`Q;`K 1tTM/2/ #bi`+ib-RN3RĜRN39X → T;2b RNGm- qX@FX- M/ GX GBm- kyyN- /TiBp2 KmHiBTH2 bm#i`+iBQM #b2/ QM+QMbi`BM2/ BM/2T2M/2Mi +QKTQM2Mi MHvbBb, :2QT?vbB+b- d9- oRĜodX→ T;2b kkJ- CX- kyRj- h?`22@/BK2MbBQMH B``2;mH` b2BbKB+ /i `2+QMbi`m+iBQM pBHQr@`MF Ki`Bt +QKTH2iBQM, :2QT?vbB+b- d3- oR3RĜoRNkX → T;2b d9JD/ƖbFB- JX- *X EQbiQp- 1X E`;?- AX JQQ`2- JX h?QKTbQM- M/ CXJBbT2H- kyRR- ii2MmiBQM Q7 7`22@bm`7+2 KmHiBTH2b #v mTf/QrM/2+QMpQHmiBQM 7Q` K`BM2 iQr2/@bi`2K2` /i, :2QT?vbB+b- de-oRkNĜoRj3X → T;2b kjJHBQmiQp- .X- JX *2iBM- M/ X qBHHbFv- kyy8- >QKQiQTv +QMiBMmiBQM 7Q`bT`b2 bB;MH `2T`2b2MiiBQM, A111 AMi2`MiBQMH *QM72`2M+2 QM+QmbiB+b- aT22+?- M/ aB;MH S`Q+2bbBM;- A111- djjĜdjeX → T;2b 93JibQM- EX >X- M/ .X *Q``B;M- kyyy-  kX8. K2i?Q/ 7Q` ii2MmiBM; 7`22bm`7+2 KmHiBTH2b #b2/ QM BMp2`b2 b+ii2`BM; b2`B2b, S`2b2Mi2/ i i?2jkM/ MMmH Pzb?Q`2 h2+?MQHQ;v *QM72`2M+2X → T;2b kkJ2iF2K2BD2`- _X X X SX X X "X- kyyk- h?2 +QmbiB+b Q7 i?2 m/BiQ`BmK Q7i?2 _QvH H#2`i >HH #27Q`2 M/ 7i2` `2/2p2HQTK2Mi, S`2b2Mi2/ i i?2S`Q+22/BM;b Q7 i?2 AMbiBimi2 Q7 +QmbiB+bX → T;2b RL22HKMB- _X- X "mKbi2BM- M/ qX _Qbb- kyy3- *QKTH2t +m`p2H2iɢ#b2//TiBp2 bm#i`+iBQM Q7 b2p2`H MQBb2 i2KTHi2b, a1: h2+?MB+H S`Q;`K1tTM/2/ #bi`+ib- je8yĜje88X → T;2b kkRjj"B#HBQ;`T?vL2Fmi- X :X- RNN3- j. bm`7+2ɢ`2Hi2/ KmHiBTH2 T`2/B+iBQM, a1: h2+?MB+HS`Q;`K 1tTM/2/ #bi`+ib- R8RRĜR8R9X → T;2b kkPH/2M#m`;- .X qX- RN3R-  +QKT`2?2MbBp2 bQHmiBQM iQ i?2 HBM2`/2+QMpQHmiBQM T`Q#H2K, :2QT?vbB+H CQm`MH AMi2`MiBQMH- e8- jjRĜj8dX→ T;2b kNPb#Q`M2- JX _X- "X S`2bM2HH- M/ "X X hm`H+?- RNNN- PM i?2 GaaP M/Aib /mH, CQm`MH P7 *QKTmiiBQMH M/ :`T?B+H aiiBbiB+b- N-jRNĜjjdX → T;2b R9RSB;2- *X *X- M/ JX X amM/2`b- RN3k- GaZ_, M H;Q`Bi?K 7Q` bT`b2HBM2` 2[miBQMb M/ bT`b2 H2bi b[m`2b, *J h`Mb+iBQMb QMJi?2KiB+H aQ7ir`2- 3- 9jĜdRX → T;2b 8RS2+Q+F- EX GX- M/ aX h`2Bi2H- RNeN- S`2/B+iBp2 /2+QMpQHmiBQM, i?2Q`v M/T`+iB+2, :2QT?vbB+b- j9- R88ĜReNX → T;2b R9_BH2v- .X *X- M/ CX 6X *H2`#Qmi- RNde- k@. KmHiBTH2 `2~2+iBQMb,:2QT?vbB+b- 9R- 8NkĜekyX → T;2b R3- ky_Q#BMbQM- 1X X- RN89- S`2/B+iBp2 /2+QKTQbBiBQM Q7 iBK2 b2`B2b rBi?TTHB+iBQMb iQ b2BbKB+ 2tTHQ`iBQM, S?. i?2bBb- JAh- *K#`B/;2- JX→ T;2b R9ěěĜ- RN8d- S`2/B+iBp2 /2+QKTQbBiBQM Q7 b2BbKB+ i`+2b, :2QT?vbB+b- kk-dedĜdd3X → T;2b k3_vm- CX oX- RN3k- .2+QKTQbBiBQM U.1*PJV TT`Q+? TTHB2/ iQ rp2 }2H/MHvbBb rBi? b2BbKB+ `2~2+iBQM `2+Q`/b, :2QT?vbB+b- 9d- 3eNĜ33jX →T;2b R8a++?B- JX .X- M/ hX CX lH`v+?- RNN8- >B;?ɢ`2bQHmiBQM p2HQ+Biv ;i?2`b M/Qzb2i bT+2 `2+QMbi`m+iBQM, :2QT?vbB+b- ey- RReNĜRRddX → T;2b R8ap2Hb- hX- EX /2 oQb- M/ CX qX /2 J;- kyRy- am`7+2 JmHiBTH2ii2MmiBQM h?`Qm;? aT`b2 AMp2`bBQM Ĝ ii2MmiBQM _2bmHib 7Q`*QKTH2t avMi?2iB+b M/ _2H .i, S`2b2Mi2/ i i?2 dkM/ 1:1*QM72`2M+2  1t?B#BiBQMX → T;2b kea+?KB/i- JX- :X 6mM;- M/ _X _QbH2b- kyyd- 6bi QTiBKBxiBQM K2i?Q/b7Q` GR `2;mH`BxiBQM,  +QKT`iBp2 bim/v M/ irQ M2r TT`Q+?2b,1*JG Ƕyd S`Q+22/BM;b Q7 i?2 R3i? 1m`QT2M +QM72`2M+2 QM J+?BM2G2`MBM;- aT`BM;2` "2`HBM >2B/2H#2`;- k3eĜkNdX → T;2b R9RRj9"B#HBQ;`T?vaTBix- aX- RNNN- Sii2`M `2+Q;MBiBQM- bTiBH T`2/B+i#BHBiv- M/ bm#i`+iBQMQ7 KmHiBTH2 2p2Mib, h?2 G2/BM; 1/;2- R3- 88Ĝ83X → T;2b kkhM2`- JX- RN3y- GQM; T2`BQ/ b2@~QQ` KmHiBTH2b M/ i?2B` bmTT`2bbBQM,:2QT?vbB+H S`QbT2+iBM;- k3- jyĜ93X → T;2b R8hM2`- JX- _X 6X PǶ.Q?2`iv- M/ 6X EQ2?H2`- RNN8- GQM; T2`BQ/ KmHiBTH2bmTT`2bbBQM #v T`2/B+iBp2 /2+QMpQHmiBQM BM i?2 t@i /QKBMHR,:2QT?vbB+H S`QbT2+iBM;- 9j- 9jjĜ9e3X → T;2b R8hvHQ`- >X GX- aX "MF- M/ CX J+*Qv- RNdN- .2+QMpQHmiBQM rBi? i?2 GRMQ`K, :2QT?vbB+b- 99- jNX → T;2b 9eh?Q`bQM- CX _X- M/ CX 6X *H2`#Qmi- RN38- o2HQ+Biv@bi+F M/ bHMi@bi+FbiQ+?biB+ BMp2`bBQM, :2QT?vbB+b- 8y- kdkdĜkd9RX → T;2b R8hB#b?B`MB- _X- RNNe- _2;`2bbBQM b?`BMF;2 M/ b2H2+iBQM pB i?2 HbbQ,CQm`MH Q7 i?2 _QvH aiiBbiB+H aQ+B2iv a2`B2b " J2i?Q/QHQ;B+H- 83-kedĜk33X → T;2b 3j- R9Rh`QTT- CX X- kyye- Cmbi `2Ht, +QMp2t T`Q;`KKBM; K2i?Q/b 7Q`B/2MiB7vBM; bT`b2 bB;MHb BM MQBb2, A111 h`Mb+iBQMb QM AM7Q`KiBQMh?2Q`v- 8k- RyjyĜRy8RX → T;2b 9dlH`v+?- hX CX- M/ *X qHF2`- RN3k- MHviB+ KBMBKmK 2Mi`QTv/2+QMpQHmiBQM, :2QT?vbB+b- 9d- RkN8X → T;2b kNpM "Q`b2H2M- _X :X- CX hX 6QFF2K- M/ SX JX pM /2M "2`;- RNNe-_2KQpH Q7 bm`7+2@`2Hi2/ rp2 T?2MQK2Měh?2 K`BM2 +b2,:2QT?vbB+b- eR- kykĜkRyX → T;2b ky- kjpM .2/2K- 1X CX- M/ .X CX o2`b+?mm`- kyy8- j. bm`7+2@`2Hi2/ KmHiBTH2T`2/B+iBQM,  bT`b2 BMp2`bBQM TT`Q+?, :2QT?vbB+b- dy- ojRX → T;2bkk- 8kpM /2M "2`;- 1X- M/ JX SX 6`B2/HM/2`- kyy3- S`Q#BM; i?2 S`2iQ 7`QMiB2`7Q` #bBb Tm`bmBi bQHmiBQMb, aAJ CQm`MH QM a+B2MiB}+ *QKTmiBM;- jR-3NyĜNRkX → T;2b tB- j8- je- 93- 8y- 8R- dy- 39- RjN- R9RěěĜ- kyRR- aT`b2 QTiBKBxiBQM rBi? H2bi@b[m`2b +QMbi`BMib, aAJCQm`MH QM PTiBKBxiBQM- kR- RkyRX → T;2b j8- R9RpM /2` "M- JX- M/ .X@hX S?K- kyy3- _Q#mbi rp2H2i 2biBKiBQM M/#HBM/ /2+QMpQHmiBQM Q7 MQBbv bm`7+2 b2BbKB+b, :2QT?vbB+b- dj- ojdX →T;2b kNRj8"B#HBQ;`T?vpM :`Q2M2biBDM- :X CX X- M/ qX _Qbb- kyRR- S`BK`v 2biBKiBQM QM P"*/i #v bT`b2 BMp2`bBQM, a1: h2+?MB+H S`Q;`K 1tTM/2/ #bi`+ib-jy- j8jRX → T;2b RkepM :`Q2M2biBDM- :X CX X- M/ .X o2`b+?mm`- kyRR- lbBM; bm`7+2 KmHiBTH2biQ 2biBKi2 T`BK`B2b #v bT`b2 BMp2`bBQM 7`QK #H2M/2/ /i,:2QT?vbB+H S`QbT2+iBM;- 8N- RyĜkjX → T;2b dy- RkepM :`Q2M2biBDM- :X CX X- M/ .X CX o2`b+?mm`- kyy3- hQr`/b  M2rTT`Q+? 7Q` T`BK`v 2biBKiBQM, a1: h2+?MB+H S`Q;`K 1tTM/2/#bi`+ib- kd- k93dX → T;2b kj- ke- j9- 98ěěĜ- kyyN- 1biBKiBM; T`BK`B2b #v bT`b2 BMp2`bBQM M/ TTHB+iBQM iQM2`@Qzb2i /i `2+QMbi`m+iBQM, :2QT?vbB+b- d9- kjĜk3X → T;2b 8- e-kj- ke- j9- je- 9R- 99- 89- d9- d8- dd- 3k- NR- RkkěěĜ- kyyN#- 1biBKiBQM Q7 T`BK`B2b M/ M2`@Qzb2i `2+QMbi`m+iBQM #vbT`b2 BMp2`bBQM, J`BM2 /i TTHB+iBQMb, :2QT?vbB+b- d9- _RRNX →T;2b ke- 8e- e3ěěĜ- kyRy- AM+Q`TQ`iBM; i?2 bQm`+2 ``v BMiQ T`BK`v 2biBKiBQM,S`2b2Mi2/ i i?2 dkM/ 1:1 *QM72`2M+2  1t?B#BiBQMX → T;2b 9Ro2`b+?mm`- .X CX- RNNR- am`7+2@`2Hi2/ KmHiBTH2 2HBKBMiBQM- M BMp2`bBQMTT`Q+?, S?. i?2bBb- hl .2H7iX → T;2b kRěěĜ- RNNk- /TiBp2 bm`7+2@`2Hi2/ KmHiBTH2 2HBKBMiBQM, :2QT?vbB+b- 8d-RReeX → T;2b kkěěĜ- kyye- a2BbKB+ KmHiBTH2 `2KQpH i2+?MB[m2b, Sbi- T`2b2Mi M/ 7mim`2,1:1 Sm#HB+iBQMbX → T;2b Rj- d8o2`b+?mm`- .X CX- M/ X CX "2`F?Qmi- RNNd- 1biBKiBQM Q7 KmHiBTH2b+ii2`BM; #v Bi2`iBp2 BMp2`bBQM- S`i AA, S`+iB+H bT2+ib M/ 2tKTH2b,:2QT?vbB+b- ek- R8NeĜReRRX → T;2b kko2`b+?mm`- .X CX- M/ _X CX S`2BM- RNNN- JmHiBTH2 `2KQpH `2bmHib 7`QK.2H7i lMBp2`bBiv, h?2 G2/BM; 1/;2- R3- 3eĜNRX → T;2b kkqM;- .X- _X a#- PX uBHKx- M/ 6X CX >2``KMM- kyy3- "v2bBM@bB;MHb2T`iBQM #v bT`bBiv T`QKQiBQM, TTHB+iBQM iQ T`BK`v@KmHiBTH2b2T`iBQM, h2+?MB+H _2TQ`i 8- l"* 1`i? M/ P+2M a+B2M+2b.2T`iK2MiX → T;2b kkRje"B#HBQ;`T?vqM;- uX- kyy9- JmHiBTH2 T`2/B+iBQM i?`Qm;? BMp2`bBQM,  7mHHv /iɢ/`Bp2M+QM+2Ti 7Q` bm`7+2ɢ`2Hi2/ KmHiBTH2 ii2MmiBQM, :2QT?vbB+b- eN-89dĜ88jX → T;2b kjqbQM- >X- 6X CX >2``KMM- M/ hX hX uX GBM- kyRR- aT`bBiv@T`QKQiBM;`2+Qp2`v 7`QK bBKmHiM2Qmb /i,  +QKT`2bbBp2 b2MbBM; TT`Q+?, a1:h2+?MB+H S`Q;`K 1tTM/2/ #bi`+ib- jy- eX → T;2b dyq2;H2BM- X "X- RNNN- JmHiBTH2 ii2MmiBQM, M Qp2`pB2r Q7 `2+2Mi /pM+2bM/ i?2 `Q/ ?2/, h?2 G2/BM; 1/;2- R3- 9yĜ99X → T;2b Rjq2;H2BM- X "X- 6X oX `ȹDQ- SX JX *`pH?Q- _X >X aiQHi- EX >X JibQM-_X hX *Qi2b- .X *Q``B;M- .X CX 6Qbi2`- aX X a?r- M/ >X w?M;- kyyj-AMp2`b2 b+ii2`BM; b2`B2b M/ b2BbKB+ 2tTHQ`iBQMX → T;2b ky- 3yq2;H2BM- X "X- 6X X :bT`QiiQ- SX JX *`pH?Q- M/ _X >X aiQHi- RNNd-M BMp2`b2@b+ii2`BM; b2`B2b K2i?Q/ 7Q` ii2MmiBM; KmHiBTH2b BM b2BbKB+`2~2+iBQM /i, :2QT?vbB+b- ek- RNd8ĜRN3NX → T;2b ky- 3yqB;;BMb- CX qX- RN33- ii2MmiBQM Q7 +QKTH2t ri2`@#QiiQK KmHiBTH2b #vrp2@2[miBQM@#b2/ T`2/B+iBQM M/ bm#i`+iBQM, :2QT?vbB+b- 8j-R8kdĜR8jNX → T;2b RNqB;;BMb- _X X- RNd3- JBMBKmK 2Mi`QTv /2+QMpQHmiBQM, :2Q2tTHQ`iBQM-Re- kRĜj8X → T;2b kNuBHKx- PX- RN3N- o2HQ+Biv@bi+F T`Q+2bbBM;, :2QT?vbB+H S`QbT2+iBM;- jd-j8dĜj3kX → T;2b R8uQm- uX GX- M/ JX Ep2?- RNNe-  `2;mH`BxiBQM TT`Q+? iQ DQBMi #Hm`B/2MiB}+iBQM M/ BK;2 `2biQ`iBQMX, A111 h`Mb+iBQMb QM AK;2S`Q+2bbBM;- 8- 9ReĜ9k3X → T;2b 9kumM- :X@sX- EX@qX *?M;- *X@CX >bB2?- M/ *X@CX GBM- kyRy- *QKT`BbQM Q7 PTiBKBxiBQM J2i?Q/b M/ aQ7ir`2 7Q` G`;2@b+H2GR@`2;mH`Bx2/ GBM2` *HbbB}+iBQM, h?2 CQm`MH Q7 J+?BM2 G2`MBM;_2b2`+?- RR- jR3jĜjkj9X → T;2b 9dwBQHFQrbFB- X JX- .X "X hvHQ`- M/ _X :X EX CQ?MbiQM- RNNN- J`BM2b2BbKB+ rp2}2H/ K2bm`2K2Mi iQ `2KQp2 b2@bm`7+2 KmHiBTH2b,:2QT?vbB+H S`QbT2+iBM;- 9d- 39RĜ3dyX → T;2b kR- kjwBQHFQrbFB- X JX- .X CX hvHQ`- M/ _X :X EX CQ?MbiQM- RNN3- JmHiBTH2rp2}2H/b, a2T`iBM; BM+B/2Mi 7`QK b+ii2`2/- mT 7`QK /QrM- M/Rjd"B#HBQ;`T?vT`BK`B2b 7`QK KmHiBTH2b, a1: h2+?MB+H S`Q;`K 1tTM/2/ #bi`+ib-R9NNĜR8ykX → T;2b kRRj3AppendixAParetoroot-findingfortheminimum ℓ1-normsolutionh?2 S`2iQ +m`p2 Q7  #bBb Tm`bmBi /2MQBbBM; U"S.LV T`Q#H2K +?`+i2`@Bx2b Bib QTiBKH bQHmiBQMbc Bi `2Hi2b Mv +?QB+2 Q7  i`;2i KBb}i σ iQ Bib+Q``2bTQM/BM; QTiBKH ℓ1@MQ`K Q#D2+iBp2X "2+mb2 i?2 "S.L T`Q#H2K Bb+QMp2t- i?Bb `2HiBQM Bb #BD2+iBp2X h?2 KQbi +QMp2MB2Mi rv iQ 2tT`2bb i?2S`2iQ +m`p2 7Q` Qm` Tm`TQb2b Bb iQ r`Bi2 i?2 QTiBKH KBb}i σ(τ) Q7  ;Bp2MQTiBKBxiBQM T`Q#H2K b  7mM+iBQM Q7 τ i?2 ℓ1@MQ`K +QMbi`BMi BKTQb2/ QMi?2 mMFMQrMX AM Qi?2` rQ`/b- r2 /2}M2σ(τ) := minimizex∥y −Ax∥2 bm#D2+i iQ ∥x∥1 ≤ τ. UXRVLQi2 i?i i?Bb 7Q`K Bb Mim`H 7Q` /2b+`B#BM; GbbQ T`Q#H2Kb- r?2`2 σ(τ)Bb i?2 QTiBKH Q#D2+iBp2 pHm2 7Q`  ;Bp2M ℓ1@MQ`K #HH +QMbi`BMi τ X a226B;m`2 XR 7Q` M BHHmbi`iBQM Q7 i?2 S`2iQ +m`p2 BM  k/ THM2 Q7 TQbbB#H2bQHmiBQMb +?`+i2`Bx2/ rBi? ∥x∥1 QM i?2 ?Q`BxQMiH tBb M/ ∥y −Ax∥2 QMi?2 p2`iB+H tBbXh?2 B/2 Q7 HQQFBM; i i?2 S`2iQ +m`p2 7Q` ℓ1@MQ`K KBMBKBxiBQM BbMQi M2rX Ai rb /2b+`B#2/ BM pM /2M "2`; M/ 6`B2/HM/2` Ukyy3V M/.m#2+?B2b 2i HX Ukyy3V b i?2 #bBb Q7 MQp2H bQHp2`b- M/ BM >2MM2M72Mi 2i HXUkyy3V b  rv iQ +?`+i2`Bx2 i?2 p`BQmb QTiBKBxiBQM `QmiBM2b 7Q` ;2QT?vb@B+H +QKT`2bbBp2 b2MbBM; T`Q#H2KbX aT2+B}+HHv- pM /2M "2`; M/ 6`B2/HM@/2` Ukyy3V T`Qp2/ i?i i?2 S`2iQ +m`p2 b /2}M2/ ?2`2 Ub  7mM+iBQM Q7 τVBb +QMp2t- /2+`2bBM;- M/ +QMiBMmQmbHv /Bz2`2MiB#H2 7Q` τ ∈ [0, τ∗]- r?2`2RjNTT2M/Bt X S`2iQ `QQi@7BM/BM; 7Q` i?2 KBMBKmK ℓ1@MQ`K bQHmiBQMkyAxk 2(0) = kyk2⌧ = ⌧⇤region of feasible solutionsregion of infeasible solutionsPareto curve⌧ := kxk1(⌧˜)(⌧)UV˜⌧k ⌧k+1g˜kg˜k+1ksolving LASSO via projected gradientU#V6B;m`2 XR AHHmbi`iBQMb Q7 i?2 S`2iQ +m`p2X SM2H UV b?Qrb i?2 S`2iQ+m`p2 BM i?2 T`K2i2` bT+2 Q7 TQbbB#H2 bQHmiBQMbX SM2H U#V /2TB+ib QM2Bi2`iBQM Q7 i?2 `QQi@}M/BM; T`Q+2/m`2 iQ }M/ i?2 KBMBKmK ℓ1 MQ`K i?i;Bp2b i?2 i`;2i 2M2`;v KBb}i σ˜Xτ∗ Bb i?2 QTiBKH Q#D2+iBp2 Q7 i?2 "S T`Q#H2K minx ∥x∥1 bXiX y = AxX 6m`@i?2`KQ`2- ;Bp2M M QTiBKH bQHmiBQM x˜ i Mv TQBMi QM i?2 S`2iQ +m`p2-i?2 bHQT2 λ Q7 i?2 S`2iQ +m`p2 i i?i TQBMi Bb ;Bp2M #v i?2 +HQb2/ 7Q`K2tT`2bbBQM λ = −∥AHr∥∞/∥r∥2- r?2`2 r Bb i?2 `2bB/mH p2+iQ` y −Ax˜- M/R9yTT2M/Bt X S`2iQ `QQi@7BM/BM; 7Q` i?2 KBMBKmK ℓ1@MQ`K bQHmiBQMi?2 ℓ∞ MQ`K Bb 2[mH iQ i?2 H`;2bi #bQHmi2 pHm2 KQM;bi HH i?2 2H2K2MibQ7 i?2 p2+iQ`Xh?2b2 T`QT2`iB2b HHQr mb iQ }M/ i?2 QTiBKH ℓ1@MQ`K τ˜ 7Q` Mv "S.LT`Q#H2K- #v mbBM; L2riQMǶb K2i?Q/ iQ bQHp2 σ(τ˜) − σ˜ = 0 UpM /2M "2`;M/ 6`B2/HM/2`- kyy3- kyRRVX _272` iQ 6B;m`2 XR# 7Q`  /2TB+iBQM Q7 i?BbT`Q+2/m`2X ai`iBM; rBi? Mv bQHmiBQM i?i HB2b QM i?2 S`2iQ +m`p2 x˜k- rBi?ℓ1@MQ`K Q7 τk M/ KBbKi+? σk UA Qp2`HQ/ i?2 Bi2`iBQM +QmMi2` k iQ F22Ti`+F Q7 i?2 L2riQMǶb K2i?Q/ Bi2`iBQMbV- r2 Bi2`i2 Qp2` bm#b2[m2Mi ;m2bb2bQ7 i?2 `QQi mbBM; i?2 `mH2 τk+1 = τk +∆τk r?2`2 ∆τk = −(σk − σ˜)/λkXAM i?2 ivTB+H +b2 r?2`2 MQBb2 rBHH +mb2 i?2 Q#b2`piBQM y iQ MQi #2 BMi?2 `M;2 Q7 A- i?2 S`2iQ +m`p2 rBHH MQi `2+? σ = 0- #mi rBHH BMbi2/ ?p2M bvKTiQi2 i i?2 H2bi@b[m`2b KBb}i σ∗ = min ∥y−Ax∥2 UpM /2M "2`;M/ 6`B2/HM/2`- kyRRVX q2 i?2`27Q`2 M22/ iQ KF2 bm`2 i?i Qm` i`;2i`2bB/mH σ˜ biBb}2b σ˜ > σ∗X  T`;KiB+ TT`Q+? iQ i?Bb +QM/BiBQM Bb iQKQMBiQ` λ−1 i?`Qm;? i?2 L2riQM Bi2`iBQMb M/ biQT i?2 BMp2`bBQM r?2M BibK;MBim/2 #2+QK2b mM`2bQM#Hv H`;2- r?B+? rQmH/ BM/B+i2 i?2 bvKTiQi2i σ∗XaBM+2 Q#iBMBM; i?2 bHQT2 BM i?2 L2riQM `QQi@}M/BM; b+?2K2 `2[mB`2bFMQrBM; M QTiBKH bQHmiBQM x˜ QM i?2 S`2iQ +m`p2 7Q` 2+? Q7 i?2 ℓ1@MQ`K`2bi`B+iBQMb τk- Bib bm++2bb ?BM;2b QM i?2 #BHBiv iQ [mB+FHv bQHp2 T`Q#H2Kb Q7i?2 7Q`Kminimizex∥y −Ax∥2 bm#D2+i iQ ∥x∥1 ≤ τ˜ . UXkVh?Bb Bb ivTB+HHv +HH2/  GbbQ T`Q#H2K- r?B+? /Bz2`2MiBi2b Bib2H7 7`QK"S.L #v M BMi2`+?M;2/ `QH2 #2ir22M i?2 Q#D2+iBp2 M/ i?2 `2;mH`Bx@BM; K2i`B+- `2bmHiBM; BM M Q#D2+iBp2 7mM+iBQM Bb +imHHv /Bz2`2MiB#H2 UhB#@b?B`MB- RNNec Pb#Q`M2 2i HX- RNNNVX "2+mb2 Q7 i?Bb /Bz2`2MiB#BHBiv- GbbQ T`Q#H2K +M #2 bQHp2/ Km+? 7bi2` i?M "S.L mbBM; T`QD2+i2/ ;`@/B2Mi K2i?Q/b UpM /2M "2`; M/ 6`B2/HM/2`- kyy3c a+?KB/i 2i HX- kyydc.m#2+?B2b 2i HX- kyy3VX 6m`i?2`KQ`2- i?2 H;Q`Bi?K mb2/ iQ bQHp2 2+?GbbQ T`Q#H2K +M #2 BMBiBHBx2/ rBi? i?2 bQHmiBQM Q7 i?2 T`2pBQmb GbbQT`Q#H2K- r?B+? /Bbi`B#mi2b i?2 #m`/2M Q7 /2b+2M/BM; i?2 KBb}i M/ 2Mbm`2bi?i 2+? GbbQ T`Q#H2K /Q2b MQi `2[mB`2 BM+`2bBM;Hv H`;2` MmK#2` Q7 Bi@2`iBQMbX  bm++2bbBp2 Q7 GbbQ T`Q#H2Kb bQHp2/ BM i?Bb rv- rBi? i?2 b2`B2bQ7 ℓ1@MQ`K `2bi`B+iBQMb τk T`2b+`B#2/ #v i?2 L2riQMǶb `QQi@}M/BM; K2i?Q/mMiBH i?2 QTiBKH τ Bb 7QmM/ 7Q` i?2 i`;2i KBb}i- 7Q`Kb  +QMiBMmiBQM@#b2/"S.L bQHp2` i?i rb BMi`Q/m+2/ BM pM /2M "2`; M/ 6`B2/HM/2` Ukyy3VXR9R

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