Multi-scale 2D acoustic full waveform inversion with high frequency impulsive source

Transcription

1 Muli-scale D acousic full waveform inversion wih high frequency impulsive source Vladimir N Zubov*, Universiy of Calgary, Calgary AB vzubov@ucalgaryca and Michael P Lamoureux, Universiy of Calgary, Calgary AB mikel@ucalgaryca and Gary F Margrave, Universiy of Calgary, Calgary AB margrave@ucalgaryca Summary The abiliy o combine ogeher differen grid scaling in full waveform inversion (FWI) is examined in he presen sudy, using synheic daa for he acousic wave equaion problem wih boundary Differen frequency impulsive sources are pu ogeher hrough domain decomposiion wih overlap o verify he abiliy of FWI sub-problems o mainain numerical sabiliy while obaining convergence hrough a Newon mehod roo search applied o misfi daa correlaed wih he forward propagaion wave Inroducion A useful mahemaical model in mahemaical physics can ofen be based on fundamenal conservaion laws of he corresponding physical problem, ensuring he preservaion of he mos imporan properies of he problem such as symmery, monooniciy and conservaion of physical laws where i is possible These properies give he advanage in furher heoreic invesigaion of he mahemaical problem and provide a solid base for sabiliy sudies and esimaions for numerical soluions using ieraive mehods of linear algebra and numerical analysis, adaped for self-adjoin and posiive definie linear problems In he presen sudy, we begin wih he acousic wave equaion problem (including boundary) wrien in he self-adjoin form, which provides numerical sabiliy for boh he forward and adjoin problem soluions using an implici facorizaion scheme This also ensures he mass conservaion propery for he medium while using numerically approximae spaial derivaives The FWI approach implemened in his sudy is based on a Newon mehod roo search applied o he inner produc of he misfi daa and forward wave insead of misfi funcion norm minimizaion usually used in similar FWI sudies (Virieux and Opero 9) Theory and/or Mehod The mahemaical model developed and sudied here is based on fundamenal resuls in wave equaion FWI derived by Taranola (984) and based on he heory of adjoin operaors A simplifying assumpion in his sudy is ha he impulsive source is considered as known; generalizing o an unknown source needs fuure research and such work is no presened here To summarize, wihou source esimaion, he mahemaical model we use in here consiss of hree key modules: Forward wave propagaion wih uncondiionally sable facorizaion Backward (in ime) wave propagaion wih facorizaion (adjoin problem) GeoConvenion 3: Inegraion

2 Newon mehod roo search for a cross-correlaion beween he forward and adjoin problem soluions A forward wave propagaion boundary problem (equaion ) is formulaed for he acousic equaion using a posiive definie self-adjoin operaor preserving hese properies wih Dirichle boundary condiions An impulsive poin source is locaed away from he air-soil boundary, deep ino he spaial area, o minimize any conribuion of he non-absorbing boundary layer The differenial equaions are given as: u u div ku f, x R,, T, ux where u represens he acousic pressure as a funcion of ime variable and space variable x, k kx represens he physical properies of he ransmiing medium which we wish o recover In he FWI, and he impulsive source funcion is given as a frequency modulaed Gaussian localized in space, f x x e sin This boundary problem is approximaed wih a 4 h order finie difference scheme in space and nd order facorizaion scheme in ime (Kalikin 978, Samarskii ): n n I I u u n n n n u f f f where and are differenial analogs of spaial derivaives in corresponding spaial direcions These marices are self-adjoin and posiive definie and he facorizaion scheme is sable for 5;5 The backwards propagaion boundary problem corresponding o (equaion ) is given as ì ïf = div( kñf) + g, x Î W Ì R, Î [,T ] í îï f ( = T) =, f ( x Î W) = where he source funcion g is an arbirary funcion in general, corresponding o he basis vecor of he inverse problem As many oher researchers do, we furher specify he source funcion o be misfi daa limied wih some window: d d g, obs d cal, Cross-correlaion beween he forward and adjoin problem soluions is compued in he radiional way as a numerical inegraion x, k u, I d (5) The firs order Newon mehod roo search (Bjoerck and Dahlquis 8) is implemened for a projecion of he cross-correlaion funcion on he spaial D curve corresponding o he boom line of he cross-correlaion being a non-zero funcion so ha x, k I x, k for x Thus, on each ieraion I P I () () (3) (4) I for each x below in D and k Jˆ (6) GeoConvenion 3: Inegraion

3 where Ĵ is a diagonal of he Jacobean xi / k j i, j I marix and P is a projecion operaor on he Examples Compuaional experimens were implemened in order o verify numerically he analyical mahemaical model on synheic daa The layer of soil in D region from (equaion ) is spli ino independen layers: a high resoluion weaher layer near he soil surface and low resoluion layer deeper in he subsurface The soluion for he weaher layer inversion problem is implemened using a higher frequency impulsive source (Fig a)) and layer below is inversed wih lower frequency impulsive source (Fig b)) Figure : Spaial grid for velociy coefficiens (a) k x in soil layer (b) Figure Spaial domain decomposiion wih corresponding impulse sources posiions (x): a) high resoluion grid for k x in weaher layer; b) low resoluion grid for x For boh he high resoluion weaher layer and he lower resoluion layer wih known exac soluion of FWI (Fig 3) and simple iniial approximaion of he velociy field (Fig 4) he full waveform inversion is implemened as an ieraive procedure he sandard way described in many papers and sudies (eg Margrave e al ) wih grid scaling playing he role of velociy field smooher Boh wave propagaion boundary problems (equaion ) and (equaion 3) are solved on he same spaial grid for each sub-problem in domain decomposiion As a resul, emporal grid discreizaion includes 45 poins bu for each paricular window, corresponding sub-grid is used Velociy field for each sub-problem is defined on a robus grid 9 wih a scaling facor 9 9 used in grid-scaling approach k GeoConvenion 3: Inegraion 3

4 Figure 3: Exac velociy field k EXACT x Figure 4: Iniial velociy field k APPRX x GeoConvenion 3: Inegraion 4

5 Figure 5 The convergence of an approximae soluion k APPRX x o x k EXACT The resuls of a numerical experimen as a FWI ieraive convergence are presened in Fig 5 The oal number of FWI ieraions is 5 including ieraions in he high frequency weaher layer FWI wihou low resoluion deeper layer FWI I means ha high frequency impulsive source FWI is used as a precondiioner for he deeper low frequency source FWI Each FWI ieraion includes hree forward problems (equaion ) numerical soluions, one backward problem (equaion 3) numerical soluion and one ieraion of Newon roo search (equaion 6) Moreover, each wave propagaion problem is solved in corresponding emporal window, which makes differen FWI ieraions have differen compuaional cos Conclusions From numerical experimens, one of which is presened in previous secion, we see ha he Newon roo search applied o forward wave and residual cross-correlaion funcion shows a good efficiency when applied o D acousic FWI wih an impulsive poin source Moreover, he P ieraive roo search (equaion 6) can be used as a good alernaive o well-sudied and widely used misfi daa norm d minimizaion L Moreover, in he D acousic FWI case or in oher cases implicily reduced o D by increasing he number of impulsive sources, Newon roo search applied o cross-correlaed forward wave and he residual numerically is expeced o provide a good convergence rae comparable wih misfi daa ieraive norm minimizaion Acknowledgemens This work is suppored by he POTSI and CREWES research consoria, heir indusrial sponsors, and agencies MPrime, NSERC and PIMS I References Bjoerck A, Dahlquis G, 8, Numerical mehods in scienific compuing Vol : SIAM Kalikin N, 978, Numerical mehods: Moscow, Nauka (in Russian) Margrave G, Ferguson R and Hogan C,, Full Waveform Inversion Using One-way Migraion and Well Calibraion: CSEG Convenion Absracs Samarskii A,, The heory of difference schemes: New York, Marcel Dekker Inc Taranola, A, 984, Inversion of seismic reflecion daa in he acousic approximaion: Geophysics, 49, Virieux, J and Opero S, 9, An overview of full-waveform inversion in exploraion geophysics: Geophysics, 74, WCC- WCC6 GeoConvenion 3: Inegraion 5