35(2), 137-145 Tgp Cht CAC KHOA HpC v £ TRAI B A T
f XAC DINH VAN TOC TRUYEN SONG DIEN TlT TRONG RADAR XUYEN DAT
BANG PHEP HIEU CHINH D Q N G
N G U Y £ N THANH VAN', VO MINH T R I E T ' ,
D>^NG HOAI TRUNG', LE VAN ANH CU'6NG', NGUYEN VAN GIANG^
E - mail: ntvanvldcf^gmail.com
' Truang Dgi hgc Khoa hgc Tu nhiin, Dgi hgc Quoc gia Thdnh pho Ho Chi Minh
^Viin Vgt ly Dia cdu, Vifn Hdm ldm Khoa hgc vd Cong nghe Viet Nam Ngiy nhan bai; 12-4-2013
LMfrdau
Radar xuyen dit (GPR) li phuong phip dja vit ly sii dyng sdng difn tir (thdng thudng trong khoing tin sd tir 10 dan 2000 MHz) [7, 8, 12] df nghien ciiu cic ciu tnic ting ndng dudi m$t dit nhu dy bio sat Id, sut lun, v5 bin dd cflng trinh ng^m dd thj, khio c6, giao thong, cic cdng trinh
\&y dyng, quin s\r,.., Sdng GPR duge phit dudi dang cic xung sdng dien tir nhd ing ten phit, lan truyin trong mdi trudng vi phin x^ 1^1 ing ten thu til cic m^t ranh gidi hoic cic ddi tugng cd thdng s6 thuin lpi cho vifc phan xg sdng difn tir [4].
Thdng s6 v§n tdc truyfn sdng difn tir d$c trung cho mdi trucmg nghiSn ciiu li van de quan tim hing diu cua ngudi khio sit ciu tnic dja chat bing phuong phip GPR. Cd nhifiu phucmg phip di vi dang dugc su dyng de xic djnh v^n tdc truyen sdng difn tir nhim ning eao dg ehinh xic trong ph6p minh giii cic di thucmg GPR nim ben dudi mit dit. Trong dd, khi phd bi^n vi don gian li sii dung v^n t6c theo bing chuin dgi difn cho mpt sd moi truimg dja chit chung lim gii trj tucmg ddi eho khu vyc dang nghidn ciiu. Tuy nhifin, sir dyng each trfin vin phii chip nhfn mft sai sd. Sai sd niy trong mft s6 trudng hgp khdng ding kS nhung voi mdi trudng dja chit cd df dSn cao thi cd thi dgt tdi 10%, d$c bift khi sii dyng tin sd ing ten thu phit thip [2, 3, 6]. Phucmg phip do difm siu chung (CMP) d^ xic djnh v|n tdc tmyln sdng difn tii die trung cho moi trudng khao sit dugc cho li chtnh xic vi phu hgp v6i tirng d6i tugng nghifn ciiu.
Tuy nhien, hau het he thdng thiet bj GPR sir dung trong khu vuc dd thj cd cae ing ten thu vi phit sdng difn tir deu dugc ghep co djnh trong mft hgp, khdng de dang tich rdi de thuc hien cic ph^p do CMP [9, 13]. Trong cdng trinh niy, chiing tdi nghien ciiu sir dyng phep hieu chinh dgng (Normal MoveOut: NMO) nhim xic dinh vin tdc truyin sdng ddi vdi so lieu do theo kieu khoang cich chung (Common Offset: CO) trong phuong phip GPR [13],
2. Ca sd toin ly eua phutrng phip 2.1. Radar xuyen ddt
Trong mdi trudng ddng nhat vi ding hudng, sdng GPR tuan theo hf phuong trinh Macxoen vi cd dang sdng phiic hay cdn ggi la phucmg Irinh Helmholtz [1, 12]. Tir hf phuong Uinh Macxoen [7, 8] ta c6 phucmg trinh sd song sau day:
I fl£ + lO}p(J (I) Trong dd, ra li tin sd gde, a li dg dan difn, e li dg thim difn va p li dg tir tham ciia mdi trudng dia chit. Ty sd gitla mat dp dong dan Jf = oE tren mat dg dong dich Jd = icoeE trong mgt mdi trudng la:
\Jj\ ae
Trong dii tin sd radar xuyen dat thi ty sd tren day ludn cd gii trj nhd hem 1 cho da phin cic logi
dit di thudng gip, vi the mdi trudng dat da cd the coi li mdi trudrng difn mdi. Phucmg trinh s6 sdng (1) cd the bieu dien qua thanh phin thyc va thanh phin io theo bilu thuc sau:
.-,9
=±7^
at fts +1 Gifja (3) Phan ao eiia sd sdng (p) thuong bifu thj hf sd pha hoic hf sd truyen, con phin thuc (a) biSu thi hf sd suy giam. Dp sau ldp da ciia sdng phing liSn quan den a:1 (4)
Budc sdng (V) va vin tflc truyin sdng (v) lien quan den |3 :
1
4^
(5)Trong dd e, li do difn tham tuong ddi; fi,. la do tir thim tuong ddi vi cd gia trj gin bing 1 cho phan Idn cic loai dat da; c la vein tdc truyen sdng difn tit trong chan khdng, Nhu vay, van tdc truyin sdng difn tir trong mdi trudng vit chat cd dp din thip phy thudc chii ylu vao do dien thim.
Kft qua nghien cuu ciia Davis vii Annan (1989) vi cua cic tic gia khic [3, 4, IS] cho mgt mdi trudng dja chat di chi ra ring d§ dan difn inh hudng rit manh din hf s6 suy giath ciia sdng radar cdn vin tdc cua sdng radar Igi phy thu$c nhif u vio do difn thim. Ddng thdri hf sd suy giim ciia sdng radar phu thu^c rit mgnh vio tin sd.
Viec tinh toin vin tdc truyen sdng bibig phuong trinh (5) cd thf dhv den sai s6 Idn do mdi trucmg dja chit Ii mdi trudng bit ddng nhit khdng ding hudng. Ph^p hifu chinh d$ng (NMO) duge sft dyng dl xic djnh vin tdc truyen sdng GPR tii s6 lifu thu dugc theo kilu do khoing cich chung nhim khic phyc hgn chf niy.
2.2. Phep hi^u chinh dpng trong thdm dd dia ch&n Phucmg phip NMO duge sii dyng dl chuyin da lifu vdi khoang cich thu - phit khic 0 thinh dil lifu vdi miy thu vi miy phit triing nhau. Tiic li, phuong phap NMO dua bilu dd thdi khoang diem sau chung cua cic sdng phan xa vl diing dudng thing t = to, dam bao cho vifc cgng d6n pha cic dao ddng cd fch [5, 15, 9].
Xel phuomg trinh truyfn sdng ddi vdi dQ lifu diem sau chung {hinh I).
Hinh 1. Saab b6 tri tuyen do diem sau chung (a); M^t cit dja chin diem siu chung (b) Khi bit dau xu ly so lifu, vi khong biet ro thi
nim cua mit phan xa ciing nhu khdng bilt chinh xic vgn Idc truyfn sdng trong mfli trudng nen df hifu chinh dpng ngudi ta xem mdi trudng mit phin xg li nim ngang, cdn van tdc truyfn sdng thi sit dyng cic sd lieu tien nghiem hoic thii sai. Ta cd phuang trinh:
(flKil-
-'-iKf-"
(6)trong d6, v(to) 14 v^in t6c thii nghidm t^i thdi gian to; X 14 Idioang cdeh gida n6 vi thu; h ]i dO sSu ciia
cdnli giti ph&n X9; to \k thfri gian sdng tnQ'ln dgc theo tia p h ^ tuy^n khi miy thu d^t triing miy phit vi triing t^i dilm giQa.
Tir cfing thiic (6) c6 thi thiy ring, iing vdi quy Iu$t v(tg) nhit djnh lugng hifu chinh dfng trfn mdi dudng ghi dao d^ng (vj tri x) thay d6i theo to.
Xft truJmg hgp m^t phin xg nim nghieng gde (p (nlu m$t phing ngang thi ip = 0) vi v^n tdc tmyln sdng trong Idp phii Ii v,.
Phuong trinh thdi khoing ddi vdi mit phing nghifng Ii:
1 2 ~
2 X COS (p (7)
tjjj li thdi gian quan sit dugc trfin lit cit dja chin cua sdng,
Trong trudng hgp niy, bilu dfl thdi khoing ciia sdng sau hifu ehinh d^ng cd dgng:
t = t,,-At (8) d day t - thdi gian sau khi hifu chinh,
Trfn CO sd cdng thiic (6, 7 vi 8), ta cd:
Thiic hi^n phdp Idiai triin chuoi c4c bi€u thuc chira can thtic v4 k^ hi0u Vosc = —— , ta dua vl dang;
cos<p
I fcos'ip ^] ! 2t;
1 I cos'p • (10)
Nhu vgy sau khi tien hinh hifu ehinh dgng so bf. Bifu dd tiidi khoang cua sdng phin xg mfl tin dugc nin thinh parabol dang [11]:
t = to+ax^+bx*
(^ ^ 1 ; IvLc v^(t„)J
b = — -
(11)
(12)
1 1
S t o U k c ' v ^ t j j Tir cic cdng thiic (11) vi (12) ed thi thiy ring, nlu v(to) > VBSC thi t > tg, dudng t nim tren dudng to (theo chilu ting cua true thdi gian).
Trudng hpp niy cdc tryc ddng pha dirpc nin chua tdi. Ngugc lai, khi v(to) < v^sc thi cic true ddng pha dugc nin qui, hay li t < t-.
Tryc ddng pha ciia sdng phin xa ehi dugc nin thing khi van toe v(to) chpn diing bing VBSC- NghTa
\k khi: v(to) - VBSC thi t = tp {hinh 2).
VI dieu kifn (6) dam bio cfng bien dp cic tin hifu cd fch n£n qui trinh tinh chinh hifu chinh dgng ve thyc chat cd the xem li qui trinh tim kiem cic quy luat v(to) vi tinh lai hifu chinh dpng theo quy luat van tflc tim dugc.
V(to)<VDSC V(to)>VDSC V(to)=VDsc
Hinh 2. Lit cit dja chin dilm sSu chung (a); Lat cit hifu chinh nin qui, v(ta) < v^sc (b):
Lit cit hifu chinh nin chira td'i, v(to) > vgsc (c), Lat cit nin thing, v(ta) = VQSC (d)
Trong trudng hgp khio sit bing hf th6ng miy kilu khoang cich chung [12, 13}. Neu dj thudng bfn GPR cd min chin thi sd lifu thu n h ^ dupc theo duoi m$t dit ed dgng dilm (kich thudc rit nhd) Ih)
tin hifu thu dugc tuong duong vdi tin hifu theo kieu CMP; vi ddi voi ing ten GPR cd min chin, khoang cich giiia ing ten phit vi thu thudng rit nhd, cd 10 cm (ing ten 500 MHz cua Hang Mali, Thyy Difn), nfn sai sd giua thoi gian phit va thu so vdi thdi gian khi khoang each ing ten phit va thu bing 0 la
khdng dang ke [13]). Khi dd, co sd IJ thuylt hifu ehinh dgng dugc kp dyng nhu nhau cho ci hai tnidng hgp dilm sau chung vi khoang cich chung.
D I cu thf hon, xft phuang trbih thdi khoang ciia sdng GPR ddi voi cich bd tri do dac khoang cich chung trong hinh 3.
IHinh 3. Bd tri 6o dac Ichoang c^ch chung (tryong hgp trong hinh ve niy thi Cikm phit S v i diem thu R gin nhu' trUng nhau) v i gian 36 GPR
Sii dyng djnh ly Pythagor ta cd:
SG =MG''+MS^o
2 J
(13)
Vdi, x: khoang each giiia vi tri phit - thu va vj tri dj vgt G tren tuyen do; v: van tdc trung binh bSn tren mit phan xa; to: thdi gian hai lin sdng truyfn doc theo MG.
Nhu vSy, ca hai phucmg trinh thdi khoang trong kieu do difm sau chung vi khoing cich chung diu phan anh cic hyperbol. Do do, phuong phap hifu chinh ddng cd the ip dung ddi voi dir lieu khoang cich chung trong phuong phip GPR
3. Nghien cu'u va ap dung 3.1. Mo hinh ly thuyit 3.1.1. Mo hinh mgt mat phdn xg
Trong phan nay se trinh bay md hinh sd lifu cho truong hpp mdt mit phan xa ngang. Day li md hinh don gian nhat va thucmg gap nhSt trong thyc te.
Ta se su dung phuang trinh (6) dl thyc hien hifu chinh trong trudng hgp nay. Dua vao gia trj vgn tdc va gii tri to ta diing chucmg trinh Matlab dl thuc hifn tinh toan gia trj At, sau do tiln hinh hieu chinh cho den khi hyperbol trd thanh dudng thing thi gii tri van tdc Iiic dd chinh la van tflc cin tim [5, 13].
140
Hinh 4 li mit cit diSm siu chung tir md hinh mdt mit phan xg ngang.
- Dudng thing 1 bieu dien sdng khdng khi.
- Dudng thing 2 bilu diln sdng dit.
- Dudng hyperbol 3 bifu dien sdng phin xg, cic thflng sd md hinh li to = 100 ns, v = 0,1 m/ns vi khoang eich x tii da li 30 m.
Thyc hifn hifu chinh vdi eie v§n tdc Ii 0,08;
0,1 vi 0,12 nVns ta dugc cic kft qua nhu trfn hinh 4b, c, d,
Theo kit qui hifu chinh ta thiy vin t6c hifu chinh diing li v = 0,1 m/ns, kit qui niy phii hgp vdi sfl lieu md hinh.
3.1.2. Md hinh ba mat phdn xa
Trong phin nay chiing tdi trinh biy mfl hinh sd lieu cho trudng hgp nhieu mit phan xg ngang.
Hinh 5 trinh biy md hinh s6 lifu ciia trudng hgp nhif u mit phan xg ngang:
- Dudng thing 1 bilu dien cho sdng khflng khi.
- Dudng thing 2 bifu diln chg sdng dit - Cae dudng hyperbol 3, 4, 5 bifu dien cho ba ranh gidi phan xa khic nhau lin lupt cd gii trj vgn tdc V| = 0,12 m/ns; V2 = 0,11 m/ns; V3 = 0,1 m/ns vi thdi gian t^ = 100 ns; toj = 150 ns; to3 = 200 ns.
Thyc hifn hifu chinh vdi cic van tdc khic nhau ta dugc cac kit qua: hinh 4b bilu diln kit qua hifu
chinh NMO vdi v^n toe v = 0,12 m/ns va thdi gian to = 100 ns, dudng l^peibol 3 dugc nin thing, cac hypeibol cdn l^i khdng dugc nin tbeo quy luat hifu chinh NMO do khac thdi gian to; tucmg ty ta cd kit
Mo Mnh mot mat pTian XM ngang
qui hifu chinh NMO cho hai hypeibol cdn Igi lan lugt la \ = 0,11 m/ns va 0,1 m/ns {hinh 4c. 4d) Nhir vgy. ket qui hifu chinh NMO hoan toan phii hcrp vdi s6 Uf u md hinh.
M o hin tt m o t m a t p h a n xa n g a n g
^-^ __—^====^^
h [an] KbadnScAcb t"4 Hinh 5, (a) Mat cit GPR cua m5 hinh nhi^u mat phan xa ngang va l<et qua hieu chinh d^ng;
(b)v = 0,12m/ns;{c}v = 0,11 m/ns, (cl)v = 0,1 m/ns
3.2. S6 lifu thyc te
3.2.1. M$t cdt GPR diim sdu chung (CMP) tgi Qugn 9, Tp. HCM
Ap dyng tinh toin vin tflc vdi sd lifu DSC dugc do dac tai QuSn 9, Tp. HCM.
Kft qui sau khi thyc hifn hifu chinh NMO cho thay van tdc mdi tnidng bfn tren mit phan xg Ii
V = 0,3 m/ns.
Day Ii v§n tdc truyin sdng trong mdi trudng khong khf. Dilu niy dugc giii diich do ing ten sii dung khi thu thap s6 lifu la 9ng ten tin sd 100 MHz khong man chin. Do dd, sdng thu dugc vdi v3n tdc 0,3 m/ns cd thi phin xg til mgt di vit nio do nim trfn mit dit mi mdi trudng tir vSt din ing ten tiiu li khdng khi {hinh 6).
•^flY-j^ ! E & E ^
•^^,
Hinh 6 (a) lyat cat GPR (CIMP) va l<lt qua hieu chinh dgng voi vgn t6c: (b): v = 0,2 m/ns; (c): v = 0,3 m/ns
3 2 2. Mgt cdt GPR (CO) tgi 192 - 194 Dinh Tien Hodng. Qugn 1. Tp. HCM
Hinh 7 xuit hifn raft hyperbol tan xa tai vj tri to
= 19 ns, Xo = 6,9 m, Thuc hien hifu chinh trong khoang tir 5,5 m den 8 m. Thay ddi gii tri van tdc cho den khi hyperbol tan xg dugc nin thing thi diy li van tdc trung binh ciia mfli tnrdng ben tren vj tri tin xg.
Kit qui sau khi thyc hifn hifu chinh cho thiy van tdc mdi trudng bfn trfn vi tri tan xg li v = 0,11 m/ns. Theo danh gii ciia chiing toi (cd kft hgp vdi thdng tin thye dia), day li tin hifu thu dugc tii sy phan xg sdng difn tir cua dudng flng dan nudc ciiu hda, sau I m, C) = 0,15 m.
3.2.3. Mgt cdt GPR (CO) tgi cdng NghTa trang Liii sl Tp. HCM
Hinh 8 xuat hifn mgt hyperbol tin xg tai vj tri to
= 60 ns, Xo = 17,5 m. Ta se thyc hifn hieu chinh trong khoing tii 5 m d£n 30 m.
Kit qui sau khi thyc hifn hifu chinh cho thiy van toe mdi tnidng bfn trfn m?it phan xg Ii v = 0,3 m/ns. Cd thi dl ding nhan thiy diy li van tflc truyen sdng trong mdi trudng khdng khi (v§n tdc truyen sdng dien tir cyc dai), nhung chiing tdi van tiep tyc hifu chinh vdi nhung vin tdc ton hon dl kiem chirng phuomg phip xu 1;^ ciia minh vi nhin thiy vdi v > 0,3 m/ns (vd ly) thi dudng cong cong xuong, Nfu theo kit qui nay, mdi trudng tinh tir mit dit dfn dj thudng li moi tnidng khdng khi, difu niy li khong thi. Dya vio thdng tin hifn trudng khio sit (hinh 9), chiing tdi di xic dinh diy li tin hifu phin xg tir c^t difn bSn tren mit dit do ing ten khdng min chin thu dugc. Cot difn cd dudng kfnh rit nhd (khoing 0,2 m) nfin vifc ip dyng ph6p hieu chinh ddng li ding tin cay.
Nhu vgy, tii vifc xic djnh vin. t6c truyin sdng ciia tin hifu phan xg, ta cd thi xic djnh chinh xic cie ddi tugng khao sit vi khong giy ra nhiing sai sdt trong qui trinh minh giai.
KlKite«<:«dti[in1 Hmh 7. (a) Mat cat GPR kfioang cach chung tuyen do 192 - 194 Dinh Tien Hoang va kit qua h:eu chinh done
(b) V = 0,09 m/ns, (c) v = 0,11 m/ns, (d) v - 0,i3m.'ns
KlKiAnj|c&cti [ m ]
Hinh 8. (a) M^t c i t GPR Ichoing each chung tai cdng NghTa Trang Liet sT, Tp HCM va l<^t qua hieu chinh dpng, (b) V = 0.28 m/ns, (c) v = 0.3 m/ns; (d) v = 0,34 m/ns
Hinh 9 Vj tri cot dien trong Ichu virc l<hio sat
3 2.4. Mat cdt GPR (CO) tgi ga Li Thj Riing - Metro 2, Tp. HCM
Hinh 10 xuat hifn hai hypeibol tan xg tgi vi hi!«
= 18 ns. Xoi = 1 m va to2 = 20 ns, Xo2 = 3,5 m. Lin lugt thuc hifn hifu chinh trong khoing tii 0,3 m din 1,5 m va 3 m ^ n 4,3 m dl xac dinh vgn toe tn^-en sdng trong mdi tmdng bfn tren di vat 1 v^ 2,
Ket qui sau khi thyc hifn hifu chinh cho thay vgn toe mdi tmdng bfn trfn mgt phin xg dj vgt 1 la Vi = 0,08 m/ns va di vat 2 la Vi = 0,075 m/ns. Sau khi dich chiQ'en Kirchhoff vdi cac van toe nay ta thay bifn dg tin hjfu ciing dat gia tn cao nhit. Kit qua tinh toan (cd ket hgp vdi thdng tin thyc dia): di
3saBijar!f'.!t<M'i.!g
•^ig^gga^^gtf^-'^^
i
vgt 1 la dudng ong cap nudc ngam, d = 0,64 m.
<1) = 0,25 m; di \at 2 la dudng dng cap difn ngim.
sau 0,83 in, 0) = 0,45 ra 4. Ket luan
- Phep hifu chinh dgng (NMO) la phuang phap xir ly so lifu dugc su dyng nhifu trong dia chin tham dd cd thf ap dyng de xac dinh\an tdc rit hifu qui trong GPR. Khi quan sat sy thay ddi trudng sdng trong phep hifu chinh dpng se cho ta cai nhin tong quit \e sy phan bo van toe truyen sdng tgi khu \'uc nghifn cim, giup giii doan chinh xac ve dg sau va kich thudc ciia cac di vgt.
- Vifc ap dyng phep hifu chinh dgng nhim xic dinh \gn tdc tru>In sdng difn tu tren ci mo hinh
Hinh 10- (a) Mat cat GPR khoing cich chung do tai ga Le Thi Rieng - IVletro 2, Tp HCIW va \<.i\ qua hieu chinh dong, (b) d6i vm di vat 1, van t6c Vi = 0,085 m/ns;
(c) a6i vol dl vit 2, van t6c Va = 0,1 m/ns
toan va du hfu thyc te, dugc thyc Men bing chuoTig trinh vilt tren giao dien Matiab diu cho kit qui dang tin cay,
- Phuong phap NMO cd the sii dyng cho ci dir lifu dilm sau chung (CMP) va khoing cich chung (CO). Dieu nay cho phep xac dinh vM toe truyin sdng ke ci khi dieu kifn khio sat hoge thilt bi khdng thugn lgi cho vifc do dgc bing kieu dilm sau chung (GPR cd man chin)
- Giii quyet dugc mgt so hgn chi trong cac phan mem kem tiieo thilt bi ciia cac nhi sin xuit La CO sd de chiing tdi hoin thanh phan mim: 'Xac dinh vgn toe truyen sdng difn hi trong GPR". D^y la phan mim cd till dgc dugc tat ci dii lifu ciia cic may GPR do cac hang khac nhau sin xuat.
Ldi c5m crn: Nhdm tic gii xin chin thinh cim on Trudng Dai hgc Qudc Gia Tp. HCM di cip kinh phi thyc hifn dl til Trong dilm cip DHQG Tp. HCM 2012 - 2013: "Nghifn cuu phuong phip xic djnh vin tdc truyfn sdng difn tii trong radar xuyfn dit ddi vdi khu vyc dd thj".
TAILISUDAN
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Proceedings of the NCST of Vietnam, vol.11 (2) pp.95-101
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[8] Jol H. M, 2009: Ground Penetrating Radar:
Theory and Applications. Oxford, UK, p.365.
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PulseEKKOlOORUN, User's Guide, Versionl.2.
Technical Manual 25.
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Nxb. Giao thflng van tai. Hi Ngi.
[12] NT. Van. L.V.A. Cuong D.H Trung, KM. Triet, H.K Tuan, 2012: Kirchhoff migration for specifying velocity model in ground penetrating radar method. Proceedings of International conference on Ground Penetrating Radar (GPR), Shanghai, China @ ISBN 978-1-4673-2663-6, pp.
419-424.
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SUIUIMARY
Estimating e l e c t r o m a g n e t i c veiocity In G r o u n d Penetrating radar by Normal MoveOut (NMO) Ground penetrating radar (GPR) is a high-resolution near-surface geophysical method which employs radio waves, typically in the 10 to 2000 MHz frequency range to map structure and features buried In the ground (or in man-made structures). Electromagnetic (EM) velocity is an essential parameter in GPR. There are many solutions to delermine EM velocity for geological medium. The ctiaracteristics of groundwave on CMP radagram for EM velocity are the best ways, but most of modern GPR systems could not easily separate transmitter and receiver antennae.
Normal MoveOut (NMO) which is frequently used in seismic prospecting is a good resolution for estimating velocity.
Studying variations of wave field in the NMO method give us an overall view about the distnbution of radar velocities, which are useful for precise interpretation the depths and sizes of objects in survey areas. This study shows some results that were done by a program from Matlab in calculating GPR velocities of some computational models and practical data by the NMO method.
