Shot-geophone migration for seismic data

Transcription

1 Shot-geophone migation fo eimic data Chi Stolk Depatment of Applied Mathematic, Univeity of Twente, The Netheland

2 ouce x Seimic data poceing eceive x h time... ubuface Contempoay pimaie only poceing: Body wave, no effect fom bounday Multiple eflection ae aumed abent lineaization No clutte Vaiable backgound 1

3 Outline Pat I Imaging in a moothly vaying backgound Phae pace (micolocal) point of view Geomety of catteing Minimal data (dimenion n) / maximal data (dimenion 2n 1) Pat II Contucting multiple image fo velocity etimation 2

4 Patial lineaization Sepaate medium velocity into two pat v(x)(1 + (x)) Backgound v(x) lowly vaying elative to wavelength Reflectivity (x) vaie on the cale of the wavelength Caual acoutic Geen function G(x, t;x ) i given by ( v(x) 2 2 t 2 ) x G(x, t;x ) = δ(x x )δ(t), (x, t) R n R +, with G = 0, t < 0. Lineaization yield petubation δg ( v(x) 2 2 t 2 x) δg(x, t;x ) = 2(x) v(x) 2 2 t G(x, t;x ) Modeled data given by convolution with the ouce wavelet f d(x,x, t) = (δg t f)(x,x, t), fo (x,x ) in an acquiition et Y, t [0, T]. Patial lineai invee poblem to etimate v (nonlinea) and (linea). 3

5 Geometical wave popagation Simultaneou localization in poition and wave vecto pace (phae pace). Conide e.g. a izeable Fouie coefficient of a plane wave component e iω(p x t) even afte muting ignal outide a neighbohood of ome point (x, t 0 ). Solution of the wave equation have the popety that high-fequency local plane wave component appoximately popagate along a ay t (X(t), P(t)), with X(t 0 ) = x, P(t 0 ) = p, given by an ODE, if p and P(t) atify the dipeion elation p = v(x) 1. In ay theoy one can ditinguih between Ray, taveltime (kinematic) thi talk Amplitude (dynamic) 4

6 Geomety of catteing Suppoe δg ha a lage contibution at (x,x, t ; ωp, ωp, ω), with p = v(x ) 1, p = v(x ) 1. What doe the Bon appxomation fomula imply about (x)? ( v(x) 2 2 t 2 x) δg(x, t;x ) = 2(x) v(x) 2 2 t G(x, t;x ) 1. Thee i a eceive ay (X (t), P (t)), detemined by X (t ) = x, P (t ) = p, and 2(x) v(x) 2 2 t G(x, t;x ) mut have a lage component omewhee along thi ay. 2. Thee i a ouce ay (X (t), P (t)), with X (0) = x, P (0) = p. Only along thi ouce ay can thee be lage contibution to G(x, t;x ). P (t ) x x,, p p X (t ) = y = X (t ) (P (t ) P (t ))= k/ ω k P (t ) 5

7 3. The ay mut inteect at ome y = X (t ) = X (t ), and 2(x) v(x) 2 mut have a lage contibution at (y;k) with k = ω(p (t ) P (t )) aound y. Snell law i atified becaue P (t ) = v(y) 1 = P (t ). Relation ( catteing elation ) detemined by the ay (y, ω(p (t ) P (t )) cood. of enegy in eflectivity (x,x, t, ωp, ωp, ω) cood. of event in data Thi i the canonical elation of the map F : d a a Fouie integal opeato (Rakeh (1988), Ten Koode et al. (1998)). 6

8 Retiction to the acquiition et To localize an event in the data with epect to wave vecto (ωp, ωp ), ufficient ampling i equied. Auming (x,x ) denely ample a ubmanifold of Y of R n 1 R n 1, then component of (p,p ) nomal to Y can not be obtained diectly. In cae of maximal acquition ((x,x ) denely ampled along the plane) the dipeion elation detemine one component of p and imila fo p, and (p,p ) i fully detemined. If Y i a lowe dimenional ubmanifold of R n 1 R n 1 then (p,p ) i not uniquely detemined. 7

9 Imaging Imaging involve the adjoint of the map d, with modification that do not affect the poition of econtucted ingulaitie. The ummation involved, lead to contuctive intefeence peciely at the eflection point y that ae poible accoding to the catteing elation. A coect image i obtained only if (x,x, t, ωp, ωp, ω) uniquely detemine (y, ω(p (t ) P (t )) (ee e.g. ef. Hanen (1991), Ten Koode et al. (1998), Nolan and Syme (1997). poible nonuniquene with ingle ouce data x x nonuniquene with maximal acquiition (thi ituation will alway be excluded) x x 8

10 Etimate a velocity: v(z) example Ambiguity between velocity and eflecto depth: Need multiple ouce eceive pai Let h = (x x )/2 be the offet coodinate. Given a velocity v = v(z) thee i a tictly inceaing function z T(z, h) elating taveltime of a eflected wave to depth of the eflecto migate aival at uface to eflecto poition in depth Semblance pinciple: Fo coect v, the poition of the eflecto image i independent of the offet h. Etablih whethe v i coect fom the migated data (i.e. without taveltime picking). u b u f a c e ouce offet 2h eflecto eceive time h T(z,h) depth h 9

11 Pat II: Multiple image fo velocity etimation Geneal appoach to deive petack migation fomula Binning baed migation cheme Shot-geophone migation Geometical acoutic analyi of hot-geophone migation Example 10

12 Geneal idea of petack migation opeato Petack migation i a linea map data d(x, t;x ) (petack) image et I(x,h) ; emblance pinciple. Contuct a an adjoint of an extended fowad opeato: (1) extend the definition of eflectivity to depend on moe patial degee of feedom, uch that when the exta degee of feedom ae peent in ome pecific way ( phyical eflectivity ), Bon modeling i ecoveed; (x) j F[v] R(x,h) Bon data F[v] (2) A petack migation opeato i given by the adjoint of the extended modeling opeato F. It output i the petack image et, which depend x and the new vaiable h. 11

13 Common offet migation Baed on ubet of data of dimenion n, d(x m h,x m + h, t), with x m = (x + x )/2 the midpoint coodinate. Ue fixed h. Take oiginal Bon fomula δg(x, t;x ) = 2 t 2 dx 2(x) v(x) 2 dτg(x, t τ;x )G(x, τ;x ). Replace 2(x)/v 2 (x) with R(x,h), the additional degee of feedom ae h, the component of ouce-eceive half-offet. Extended common offet modeling opeato F co [v]: F co [v]r(x, t;x ) = u(x, t;x ), with u(x m +h, t;x m h) = 2 t 2 dx R(x,h) dτg(x, t τ;x m +h)g(x, τ;x m h). If R(x,h) = 2(x)/v 2 (x) i actually independent of h, then the output u(x, t;x ) equal that fom the oiginal Bon fowad modeling. j : (x) R(x,h) = 2(x)/v 2 (x), 12

14 Common offet migation F co[v]d(x,h) = I co (x,h), I co (x,h) = dx m dt 2 d t 2(x m + h, t;x m h) dτg(x, t τ;x m + h)g(x, τ;x m h). Fo an implementation ue that G(y, t;x) j (...) δ(t T (j) (y,x)), whee T (j) (x,y) i the multivalued tavel time function and (...) denote amplitude and filte. Then you get a Kichhoff migation fomula I co (x,h) = j,k dx m (...) d(x,x, T (j) (x,x m h) + T (k) (x,x m + h)). 13

15 Example with a focuing len Medium velocity 0 x (km) Ray and wavefont how multipathing and cautic. 0 x km Data (ingle hot) x(km) z (km) 1 z km 1 t() v (km/)

16 Image of ynthetic data A ingle image with ome contibuting ay 1.6 offet x km 0 1 tavel time #, 1,1 3,3 x ,1 1,2 3,1 z km 1 3, Violation of emblance pinciple fo common offet migation (pictue fom Stolk and Syme (2004)). 15

17 Shot-geophone migation Replace 2(x)/v 2 (x) by R(x,h) whee h i the depth (half)offet, which ha nothing to do with the ouce-eceive half-offet 0.5(x x )! The hot-geophone modeling opeato F g [v] i given by F g [v]r(x, t;x ) = u(x, t;x ), with u(x, t;x ) = 2 t 2 dx dh R(x,h) dτg(x+h, t τ;x )G(x h, τ;x ). The field u(x, t;x ) i identical to δg(x, t;x ) when R(x,h) = 2(x) v 2 (x) δ(h). Bon modeling i hot-geophone modeling following the mapping j : (x) 2(x) v 2 (x) δ(h). 16

18 Shot-geophone migation: F g[v]d(x,h) = I g (x,h), I g (x,h) = dx dx dt 2 d t 2(x, t;x ) dτ G(x + h, t τ;x )G(x h, τ;x ). Note that in the above equation, all input vaiable ae integated to poduce the value at each output vecto: the computation i not block diagonal in h, in contat to the common offet opeato. Intoduce unken ouce and eceive coodinate x = x + h, x = x h, and the ouce-eceive petack image Ī g by Ī g ( x, x ) = I(( x + x )/2, ( x x )/2). 17

19 Kinematic elation Suppoe Y i of maximal dimenion. Conide enegy localized in the data poition/fouie pace at (x,x, t, ωp, ωp, ω). Then x mut be on the ay detemined by x,p. Then x mut be on the ay detemined by x,p. The time t mut be the um of the time x x and x x. Reflecto location i not yet uniquely detemined: Only the total taveltime i pecified by the event! P (0) P (t +t ) x,p x, p X (t ), P (t ) X (t ), P (t ) = x, k / ω t + t = t t + t = t X (t ), P (t ) X (t ), P (t ) = x, k /ω 18

20 Retiction to hoizontal offet Unique eflecto location if (Stolk and De Hoop) ubuface offet ae eticted to hoizontal (h z = 0); i.e. conide I g,z (x, h x, h y ) = I g (x, h x, h y,0) ay (eithe ouce o eceive) caying ignificant enegy ae nowhee hoizontal, i.e. P,z > 0, P,z < 0 thoughout the popagation; x, p x, p event in the data detemine full (foudimenional) lowne P,P, i.e. thee i ufficient ouce and eceive coveage. P (t ) X (t ) = X (t ) k P (t ) t + t = t z = z Semblance pinciple: enegy in I g (x, h x, h y,0) i focued at h x = h y = 0. P (t ) P (t ) k / ω 19

21 Convet I g,z to function of x and angle (1) via Radon tanfom in offet and depth (Sava and Fomel,2003) Define angle migated image A z (x, y, z, p x, p y ) = dh x dh y I g,z (x, y, z + p x h x + p y h y, h x, h y ), in which p x and p y ae the x and y component of offet ay paamete. 0 h angle depth Radon tanfom depth of intecept (2) Radon tanfom in offet and time (De Buin 1990, Pucha et al 1999). Omit the technical detail about thi. 20

22 Example Compae Kichhoff angle migation: Example fom Stolk and Syme, 2004 Wave equation angle migation: Fom Stolk, De Hoop and Syme (pepint 2005) Computation: Double-quae-oot appoach uing genealized ceen (Le Roueau and De Hoop (2001)), thank A. E. Malcolm fo help. 21

23 Example 1: Len velocity model ove flat eflecto: 0 x (km) z (km) v (km/) DSR image of data len velocity model, flat eflecto: x(km) z(km) 2 22

24 Synthetic data, hot ecod at hot location -500 m. 3.5 x(km) Ray and wavefont how multipathing and cautic. 0 x km t() 4.5 z km

25 Len model, common image point gathe obtained with the Kichhoff angle tanfom at x m = 300 m. Len model, common image point gathe obtained with the wave-equation angle tanfom (ight) at x m = 300 m. angle(deg) angle (deg) z(km) 2.0 z (km)

26 Example 2: Mamoui deived model. 0 x km km 1 z km km DSR image

27 1.8 eceive poition km time Mamoui deived model, hot ecod at hot location 7500 m. x km z km 1 2 Ray oiginating fom the point (6.2, 2.4) (in km). 26

28 Mamoui deived model: common image point gathe at 6200 m. Kichhoff angle tanfom Wave equation angle tanfom angle deg angle (deg) z km 2.4 z (km)

29 Dicuion The two diffeent petack migation method have vey diffeent behavio in a backgound medium with multipathing: Migation of data ubet (binwie migation): Multipathing lead to kinematic image atifact. Thee wee etablihed in example by both geometical analyi and computation of the image. Shot-geophone migation: Geometical analyi how the abence of atifact unde aumption of downgoing ay, and complete acquiition coveage. Confimed in image of the above mentioned example. Thi method i theefoe moe uitable fo velocity analyi in complex media. In cae of maine data 3-D data an aumption i needed fo the medium vaiation in the paely ampled diection. 28

30 Some efeence Main ef fo thi talk: C. C. Stolk, M. V. de Hoop and W. W. Syme, Kinematic of hot-geophone migation, pepint, tolkcc. Othe ef.: Ten Koode, A. P. E., Smit, D. J. and Vedel, A. R., A micolocal analyi of migation, Wave Motion, 28 (1998), pp Nolan, C. J., and Syme, W. W., Global olution of a lineaized invee poblem fo the wave equation: Comm. P. D. E., 22 (1997), pp Stolk, C. C., and Syme, W. W., Kinematic atifact in petack depth migation: Geophyic, 69 (2004), pp C. C. Stolk and M. V. de Hoop, Modeling of eimic data in the downwad continuation appoach, SIAM Jounal on Applied Mathematic 65 (2005), no. 4, pp C. C. Stolk and M. V. de Hoop, Seimic invee catteing in the downwad continuation appoach, pepint. 29