PERFORMANCE MEASURES FOR SPARSE SPIKE INVERSION VS. BASIS PURSUIT INVERSION

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1 4 IEEE 8-th Convention of Electrical and Electronic Engineer in Irael PERFORMANCE MEASURES FOR SPARSE SPIKE INVERSION VS BASIS PURSUIT INVERSION Igal Rozenberg Technion IIT Haifa, Irael Irael Cohen Technion IIT Haifa, Irael Anthony Vailiou GeoEnergy, Inc Houton TX, USA Abtract Sare Sike Inverion ( and Bai Puruit Inverion ( are two method for eimic inverion which utilize are inverion technique A LARS-LASSO olver i ued for comarion of their erformance We rooe the ue of the F-meaure to better evaluate the method caability to correctly identify reflection from layer' boundarie We alo how that flexible l enalization in the LASSO olution hold otential for imroving erformance Keyword eimic inverion, are ike inverion, bai uruit inverion INTRODUCTION In the eimic inverion roblem we aim to find the ground layer' tructure from meaurement of acoutic echoe of a given, or etimated, ignal A hort eimic ule i tranmitted from the earth urface A enor array deloyed on the ground receive the reflected ule from the interface between ground layer with different imedance Following ome imlifying aumtion and a re-roceing tage, the data can be regarded a a erie of one dimenional trace In thi D forward model ([8],[7], a eimic wavelet i convolved with the reflectivity erie to roduce the eimic trace The reflectivity attern i aumed to be are a only boundarie between adjacent layer may caue a reflection of the eimic wave It i deirable to erform a de-convolution of the eimic trace to obtain etimate of the reflectivity attern A noted in the literature, the band limited eimic wavelet imoe limitation on the accuracy of the inverion rocee; ecifically, high frequencie comonent of the reflected wave may be more ucetible to noie or may not be retrievable at all It wa uggeted to ue inverion technique which inherently exloit the arity of the reflection channel In thi aer, we examine the Sare Sike Inverion ( and the Bai Puruit Inverion ( [] Both method ue a dictionary of exected ignal form, baed on the etimated eimic wavelet The olution for both method i baed on the LASSO convex relaxation of the claical are inverion roblem We comare the erformance of thee two method under variou cenario We ugget the F-meaure, a fidelity criterion additional to the correlation, to evaluate the algorithm erformance We alo refer to the arity romoting l-enalty term of the LASSO We how that a variable election of the enalty for each ecific trace can imrove the reult comared to a fixed enalty even for fixed SNR cenario In addition, we oberve that the election of the dictionary reolution for the inverion roce, can greatly affect the algorithm' erformance Latly, baed on our oberved reult we ugget further reearch direction The aer i organized a follow In Section, we introduce the mathematical model and the mathematical aroach for the olution In Section 3, we refer to everal imlementation iue in the are olution aroach In Section 4, we introduce the imulation reult and fidelity criteria In Section 5, we demontrate exerimental reult MODELING AND SOLUTION APPROACHES We adot the -D "forward model" alo utilized in [] ( ( t = w( t r( t + n( t According to thi model, the earth tructure i laminated in lanar horizontal layer Each uch layer i aumed to have uniform characteritic or "imedance" An acoutic ignal w( t traveling within thi medium will exhibit reflection at the boundarie of uch layer with different imedance Thu, the reflectivity attern r( t encaulate the deired data on the ground' tructure In tyical cenario, r( t i a are ignal The eimic wavelet w( t i aumed to be known Practically it i etimated in a re-roceing tage The meaured ignal i the convolution of the eimic wavelet with the reflectivity attern Random additive noie with uoedly known tatitic often contaminate the meaurement The final meaurement ( t i uually referred to a the eimic trace The objective of the inverion i to find the reflectivity r( t from the eimic trace ( t The eimic wavelet w( t i band limited in it nature Hence, even in the abence of noie one can hoe to recover r( t u to the bandwidth reolution w( t ermit In ractice thi tranlate to a minimal layer thickne below which we do not exect to detect In thi aer, we examine the erformance of two technique aimed to recover the reflectivity attern, namely and [] In the following, we briefly reent the model and the olution aroach We refer the reader to [] for further detail Both the and rely on the knowledge of the eimic wavelet w( t A ( imlie, the eimic trace conit of a um of w( t and it time hift, according to the non-zero reflector in r( t After time dicretization, and an addition of random noie, ( take the form of:

2 = W r + n ( N N M M N N M In the inverion method, W N M R, alo known a the Dictionary, i the convolution matrix formed column by column by amle' hift of the eimic wavelet w[ k ] The j- th column of W can be een a the meaured reone to a N M δ j clean "reflection imule" at time j: r [ k] = δ[ k j] N i the number of meaurement in the eimic trace M i the dicretized oible location of layer' boundarie The method i aimed to cature thinner layer A uch, Zhang and Catagna [] rooe to contruct the Dictionary W a even and odd wedge reflectivity air Eentially N M the Dictionary i comried of a dicretized ( t= k T verion of the ignal: A ( t, m, n, T = w( t* δ ( t m T + δ ( t m T n T (3 e [ ] for the even atom, and imilarly A ( t, m, n, T = w( t* δ ( t m T δ ( t m T n T (4 o [ ] for the odd atom The indexe m,n aume all the legitimate value The inverion roblem of finding r M from the noiy meaurement N i formulated a (5 min M N N M M r ubject to W r < ε The convex relaxation of (5 after an additional Lagrangian relaxation of the contraint i given by: (6 min N WN M rm r M r M + The roblem formulated in the form of (6 i named "LASSO" The ue of l enalty in imilar uort recovery roblem roved to romote arity (ee [], [3] of the olution r M 3 SIMULATION SETTINGS Both the and method were imulated We focued on erformance evaluation of the and method for inverion on ynthetic data under variou etting In thi ection, we decribe the following aect of the imulation environment: (i Synthetic generation of the reflection channel and the eimic wavelet; (ii Generation of the eimic trace; (iii The dictionary; (iv LARS-LASSO olver We note that we were aided by [] for chooing ome of the imulation arameter I Synthetic generation of the reflection channel the eimic wavelet The imulation wa erformed in two oerational amling rate of 5Hz and 66667Hz Mot reult hown were taken from the 5Hz etting The Ricker wavelet of 4Hz bandwidth erved a the eimic wavelet It amled verion with L = round(5 ( F / 5 amle i denoted by W w[ k] wa generated with the aid of the MATLAB function =ricker(nt,freq,dt,nwno taering wa ued and o nw wa et to w[ k ] The reflectivity attern maximal delay wa et to 5ec Hence, For the aforementioned amling rate F, the reflection attern r[ k ] tranlate into a 5 ta and 84 ta filter, reectively The channel coefficient were generated in air with a fixed but cutomizable delay D Thi acing between adjacent reflection i a crucial factor in the inverion algorithm Hence we would rather control thi acing rather than etting it at random to better evaluate the method' erformance Another cutomizable arameter i the channel' arity ( The delay between one air to the next adjacent one wa randomly choen a D + ( where ( ~ Geom( When we quetion the effect of the different amling rate, the reflection channel were yntheized in the low amling rate and u-converted to the higher rate by adding zero between amle Thi cheme allow a fair comarion of the two a the number of reflection i ket contant Latly, the magnitude of each reflection wa drawn indeendently from a uniform ditribution in the range [-, ] II Generation of the eimic trace Once the eimic wavelet and the reflectivity attern are et, the generation of the eimic trace i rather traightforward Yet we want to focu attention on ome fine detail The eimic trace i generated by convolving the eimic wavelet w[ k ] with the reflectivity attern r[ k ] Then, an additive n[ k ] IID zero-mean Gauian noie with re-ecified variance matrix, ay σ I,i added to obtain the eimic n LT LT trace [ k ] The variance i choen to reach a deired SNR: (7 [ k] = w[ k]* r[ k] + n[ k] Since we are dealing with are inverion technique, it i not only the SNR that matter but alo the ditribution and magnitude of the reflectivity coefficient r[ k ] We take a tandard aroach and et the noiele reone to have a unit norm uch that III w[ k]* r[ k] Then, the noie i generated E n[ k] = SNR The Dictionary The Dictionary i yntheized according to the guideline exlained in the reviou ection We kee in mind that r[ k ] i an L length vector, the eimic wavelet w[ k ] i an δ j length vector We denote by r [ k] = δ[ k j] the L length vector with a ingle non-zero unit value at k = j Then, The dictionary W j-th column i imly given N M δ j by: w[ k]* r [ k] The dictionary W Nb Mb ynthei i a bit more involved a it hould include all the oible odd and even air reone to the eimic wavelet a imlied by (3 and (4 Let L be the channel length, then the total number of dictionary atom Nb i given L W

3 L by Nb= ( L l + L l= The firt term correond to all even (and odd air deloyment over the channel; For a air with it early reflection at time index l, we identify L l ( + oible late reflection comanion The econd term correond to the ingle reflection cenario where the late air comanion i outide the coe Thi econd art i, a a matter of fact, a dulication of the dictionary Practically, for the ake of reducing comutation time, we yntheized the dictionary by taking only reflection air with acing lower or equal to ta We note that the hift-invariance roerty of atom' grou in the dictionary could be utilized to reduce comlexity of the inverion algorithm IV LARS-LASSO olver of M The LARS algorithm [4] can be adjuted to olve the LASSO (6 for the whole ath of The LARS method alternately augment or, at time, dro, atom from the active uort r Uniquene of the olution i guaranteed by convexity [5, 3] In our imulation, we ued the SaSM toolbox [6] for both the and inverion method We call the function 'laom' with ' X ' being either the or dictionary, ' y ' the eimic trace and ' to= ' o that we get the whole ath The function ha two outut - ' b ' and 'info' The outut ' b ' i a matrix of ize L P where L i the number of reflection coefficient (5 P i the number of junction oint 3 The t column of 'b' i alway the zero vector correonding to the olution The 'laom' function call an internal function 'larenm' We made the following modification in 'larenm': (Larenm-l36 maxste = *maxvariable; (Larenm-l3 Adding to the while loo a lower limit criteria to the reidual' norm r > e 6 3 (Larenm-l63 The error meage in cae of 'no oitive direction' i relaced by a warning and a 'break' command 4 (Larenm The unbiaed LS olution baed on the uort of 'b' i calculated and given a outut of 'larm' (ee [3], The firt three modification were needed to enable calculation of a ignificant art of the route while maintaining numerical tability Thee change became inevitable in the highly correlated dictionary The ue of the unbiaed LS olution gave a mall imrovement of the imulation reult The above modification both contribute to the tability of the imulation and reduce the execution runtime [b info]=lao(x, y, to, toreath, verboe 4 FIDELITY CRITERIA AND EXPERIMENTAL RESULTS Baed on the aforementioned etting, we evaluated the erformance of the and method on yntheized random channel with random noie We imulated everal cenario with different SNR, channel' arity, and acing between reflected air D We alo how the oible imortance of varying imulation amling rate on the inverion erformance Simulation reult for variou trilet (SNR,, D are given below under different etting For each cenario, we ran tet Two fidelity criteria, ometime referred to a erformance meaure, were ued to evaluate the erformance in the different cenario The firt criterion, alo adoted by [], i the average correlation between the actual reflectivity r[ n ] and it etimate rˆ[ n ]: (8 ρ corr < r[ n], rˆ [ n] > = r[ n] rˆ [ n] The econd fidelity criterion offer a qualitative meaure of the accuracy of the etimated layer' tructure Ultimately, one wihe to identify the location of the interface between adjacent layer; In the mathematical model (, thi tranlate into a correct identification of the uort of r[ n ] Uing the tandard definition of reciion and recall, we ue a modified verion of the "F meaure" to etimate the erformance of the uort recovery: Pr eciion Recall TP F = = ( Pr eciion+ Re call TP+ ( FP+ FN The F meaure take a value in the range F A the F-meaure relie on dichotomic value of the reflection coefficient zero or nonzero, ome threholding i neceary to mitigate the effect of noie and even numerical enitivitie Thi baic, yet effective ot-roceing, greatly imrove the active uort detection The minimum value γ we elected for a reflection coefficient to be conidered active wa et to: σ (9 γ = min + 5 σ n, σ n L For arity and ignal variance σ, the average reflection coefficient amlitude for non time overlaing reflection σ would be O In ractice, the high correlation between adjacent atom, motly oberved in mall D L cenario, may end u in maller amlitude The exected contribution of the noie to a ecific reflection coefficient i O σ Thu, a reaonable threhold for reflection coeffi- ( n 3 "Junction" where new atom i augmented or an active atom i droed

4 Amlitude Amlitude time cient amlitude may take the form: where different D Reflectivity attern D Seimic Trace (SNR,,D=(dB,6,6@5Hz time Figure - Synthetic reflection channel and it correonding noiy eimic trace lambda = 5, Corr: lambda = 3, Corr: lambda = 4, Corr: lambda = 96e-4, Corr: Figure - Recontructed reflection channel Coefficient for different value (threholded coefficient σ A + B σ L election of A, B reflect a tradeoff between oible midetection and oible fale detection The firt term in γ, inherently introduce midetection even in the noiele cae; hence, for cenario with very high SNR, a limit of σ n wa et to ignificantly reduce uch midetection n Correlation and F-meaure in Inverion SNR= db, =6, ReflectionSacing= 6mec - Fmeaure 5Hz Correlation 5Hz Fmeaure 6667Hz Correlation 6667Hz - In the firt imulation etting we evaluate the correlation factor and F-meaure by averaging reult for a ecific (logarithmic et of 4 Table how imulation reult in thi etting For each trilet we get an average erformance for each correonding The value in the table are the value which correond to the maximal erformance meaure extracted for each cenario The correlation meaure i alo comared to an oracle cenario where the actual uort i given and LS etimation i alied In Figure 3, the erformance of the two fidelity meaure are reented for a whole et of for a given cenario [trilet (SNR,, D ] During the above imulation it became aarent that the maximal correlation or mot accurate uort recovery i achievable at different value of for different channel' inverion Thi can be accounted for the different number of reflection, the acing between them, the varying reflection coefficient' magnitude and noie Following thi obervation, each of the two fidelity criteria wa evaluated under an additional etting In the econd imulation etting, whoe reult are artially given in Table, we regiter for each ecific channel inverion the actual value of for which the erformance meaure actually achieve their maximal value Interolation of the meaured erformance to a fixed grid of - then follow When reeated for variou channel, we get joint ditribution: for ( ρ max, ρmax and for ( F max, Fmax Thi econd etting i alied only for the low SNR= db cenario, ince for high SNR cenario we already oberve almot er- 4 Average erformance meaure are evaluated for on a fixed grid oint N trial -3 lambda Correlation and F-meaure in Inverion SNR= db, =6, ReflectionSacing= 6mec - - lambda -3 for which we have a minimum tatitic from = 5 different run -4 Fmeaure 5Hz Correlation 5Hz Fmeaure 6667Hz Correlation 6667Hz Figure 3 Comarion of and method - Average erformance for fixed value and run -4

5 fect recovery for ome fixed We alo kee in mind that in ractice the bet matching i not acceible but intead a mart toing criterion for the LARS-LASSO hould be alied to maximize erformance In the coe of thi aer we do not reent the whole reult of the econd etting imulation Yet we reort a conitent imrovement in the inverion erformance where the variance between different run i much maller than the imrovement rate In trying to undertand the artial ga in erformance between the reorted reult in [] and our imulation reult another tet wa erformed Thi tet involve a change in the imulation amling rate and hence dictionary atom and the acing between them Figure 3 demontrate that the inverion erformance i quite ubtantially influenced by the oerational amling rate For the ame SNR and reflectance attern tatitic, it i oberved that a lower amling rate in the order of twice of the eimic wavelet Nyquit rate, erform better than the denely amled one We ee that both and "uffer" from over-amled dictionarie but i more enitive We believe that the amling rate and dictionary atom deend not only on the eimic wavelet bandwidth but alo on the meaurement SNR and the inverion reolution we aim to achieve 5 DISCUON AND FURTHER RESEARCH The reult above reveal everal intereting aect of the are channel inverion method Alo, they give rie to further reearch direction From Table we ee that for high enough SNR, the recontruction in both method i erfect or very cloe to that Thi reult even hold true for air acing of 3 ta (at 5Hz t = 3 = 6mec which i below the exected earation 5 caabilitie of our 4Hz eimic wavelet 6 6 t = 97mec π f = π 4 = [] Alo, following the obervation in the reviou ection, the election of the dictionary hould be conidered o a to maximize erformance for a given SNR and eimic wavelet Another intereting obervation which i alo aarent i the rather trong correondence between the - for which the correlation and the F-meaure take their maximal value in the noiy low SNR cenario In the high SNR cae, the eak i quite flat o the actual maximum may not be of great ignificance It can be alo oberved that both fidelity criteria how better recovery for more are reflection attern, higher air acing and higher SNR Our reult alo indicate better erformance of the technique although with roer dictionary atom election thee difference eem to become ignificantly maller The reult above encourage a further invetigation of are inverion technique and alo a toing criterion for the LARS-LASSO algorithm which may be data and SNR deendent Thi ha a otential for imroving the inverion erformance comared to the fixed cae It i alo deirable to further examine the erformance of the Table - Fixed Lambda average Performance (F=5Hz Max Correlation ρ Max F-meaure (SNR,, D F=5Hz max ρ, max E[ ρ ] max F method comared to the In that aect, it may be beneficial to ue tructured are inverion technique We aw that different in (6 may end u in very different channel Smart toing criteria may alo be crucial in the roce for electing the bet channel etimate In that aect, erforming wie threholding or ot-roceing of the raw reflection etimate, baed on robabilitic aumtion, bulk of adjacent channel or ome a-riori knowledge, may how imroved erformance REFERENCES Oracle max E F, [ ] [] RZhang and JCatagna, "Seimic are-layer reflectivity inverion uing bai uruit decomoition", Geohyic vol76 no6, Nov-Dec [] SS Chen, DL Donoho et al, "Atomic decomoition by by bai uruit: SIAM Rev, 43(, 9 59 [3] MElad, "Sare and Redundant Rereentation", Sring [4] TEfrom, THatie et al, "Leat Angle Regreion", Annal of Statitic 4, Vol 3, No, [5] RJTibhirani, "The LASSO Problem and Uniquene", Electronic Journal of tatitic Volume 7 (3, [6] SaSM- Matlab Toolbox for erforming are regreion, htt://wwwimmdtudk/roject/am/ [7] AHeimer, ICohen, "Multichannel blond eimic deconvolution uing dyanic rogramming", Journal of Signal Proceing Volume Iue 7, Volume 7 (3, [8] AJ Berkhout, The eimic method in the earch for oil andga: current technique and future develoment, ProcIEEE 74 (8 (Augut ρ max (db,4,3 (,76 (39, (,59 (49,38 (db,4,6 (7,95 (,7 996 (45,8 (9,58 (db,6,6 (45,93 (,7 995 (49,77 (,57 (8dB,4,3 (47e-6, (56e-8,99 (5e-6, (6,46 (8dB,4,6 (3e-5, (37e-8,99 (68e-7, (3,54 Table - Adative Lambda Average Performance (F=5Hz Max Correlation ρ Max F-meaure (SNR,, D F=5Hz max ρ, max E [ ρ ] max F [ ] max E F (db,4,3 (36,83 (7,6 (7,68 (6,5 (db,4,6 (,97 (7,8 (97,85 (,69 (db,6,3 (3,75 (5,56 (9,6 (98,48