Adaptation of the iterative Marchenko scheme for imperfectly sampled data
AAdaptation of the iterative Marchenko scheme for imperfectlysampled data
Johno van IJsseldijk & Kees Wapenaar
Delft University of Technology, Department of Geoscience and Engineering, Delft, TheNetherlands
May 7, 2020
Abstract.
The Marchenko method retrieves the responses to virtual sources in the Earth’s subsurfacefrom reflection data at the surface, accounting for all orders of multiple reflections. The method is basedon two integral representations for focusing- and Green’s functions. In discretized form, these integrals arerepresented by finite summations over the acquisition geometry. Consequently, the method requires idealgeometries of regularly sampled and co-located sources and receivers. Recently new representations werederived, which handle imperfectly sampled data. These new representations use point-spread functions(PSFs) that reconstruct results as if they were acquired using a perfect geometry. Here, the iterativeMarchenko scheme is adapted, using these new representations, to account for imperfect sampling. Thisnew methodology is tested on a 2D numerical example. The results show clear improvement between theproposed scheme and the standard iterative scheme. By removing the requirement for perfect geometries,the Marchenko method can be more widely applied to field data.
Introduction
Seismic surveys are generally concerned with targetsin the Earth’s subsurface. However, structures in theoverburden can distort the response of deeper targets.Ideally, all overburden structures and their multiplereflections should entirely be removed from the data,leaving only the response of the desired deeper tar-gets. This can be achieved by redatuming the re-flection response measured at the surface to a newdatum plane below the overburden. The data-drivenMarchenko method allows for the placement of virtualsources anywhere inside the subsurface, while account-ing for all orders of multiples of the overburden (Brog-gini et al., 2012; Wapenaar et al., 2014; Slob et al.,2014). Thereafter, the receivers can be moved to thesame datum plane by a multidimensional deconvolu-tion. Thus, Marchenko redatuming effectively shiftsthe response from the surface to a new datum insidethe medium, and fully removes all interactions of theshallower structures.Although the method has been successfully appliedto real data (e.g. Ravasi et al., 2016; Staring et al.,2018), several constraints still limit the usefulness ofthe method. Marchenko redatuming is based on twointegral representations. These coupled equations canbe solved by direct inversion (van der Neut et al., 2015)or by iterative substitution (Thorbecke et al., 2017).In practice, the infinite integrals are replaced by sum-mations over the finite acquisition geometry. This re-quires regularly sampled and collocated sources andreceivers in order to retrieve proper, uncontaminatedresponses. On the contrary, non-perfect geometriescan have a significant effect on the Marchenko results (Peng et al., 2019; Staring and Wapenaar, 2019). Mostauthors, therefore, assume ideal acquisition geometrieswhen using the Marchenko method, avoiding the limi-tations arising from imperfect sampling. However, thisrestriction should ideally be relaxed or even removed,allowing for broader application of the method on fielddata.Peng and Vasconcelos (2019) consider the effects ofdifferent sub-sampling and integration scenarios. Twomain effects are identified. First, when the sub-samplingand integration occur over the same dimension, thefocusing- and Green’s functions get distorted but re-main well-sampled. Second, in the situation of sub-sampling and integration over different dimensions,the focusing- and Green’s functions are accurate forthe non-zero traces but contain spatial gaps. In thecase of irregular sampling, the second effect can partlybe removed by using a sparse inversion of the Marchenkoequations, outputting well-sampled focus functions andsub-sampled Green’s functions (Ravasi, 2017; Haindlet al., 2018). On the other hand, Wapenaar and vanIJsseldijk (2020) introduce new representations for focusing-and Green’s functions, that are distorted by imperfectsampling and integration over the same dimension.These representations involve a multidimensional de-convolution with novel point-spread functions (PSFs)to deblur the distorted focusing- and Green’s func-tions. These representations are then verified on an-alytically modeled focusing functions, that have beenderived from one-way wave-field propagators and scat-tering coefficients. However, in real scenarios thesefunctions are unavailable and have to be derived fromthe coupled Marchenko equations.In this paper we explore how we can integrate the a r X i v : . [ phy s i c s . g e o - ph ] A p r new representations for irregularly sampled data intothe iterative Marchenko scheme. First, the theory ofdeblurring the Marchenko equations with PSFs is re-viewed. Next, the paper discusses the required changesto apply PSFs in the iterative scheme. Then, we presentan altered version of the iterative scheme, that al-lows for imperfectly sampled data. Finally, the per-formance of the newly developed scheme is tested onnumerical examples. Marchenko equations
This section reviews briefly the theory of the Marchenkoequations, for a more elaborate derivation the readeris referred to Wapenaar et al. (2014) and Slob et al.(2014). As starting point, imagine an inhomogeneouslossless subsurface bounded by transparent acquisitionsurface S . The reflection response at this surface isgiven by R ( x R , x S , t ), with x R and x S the receiver andsource positions, respectively, and t denotes the time.We define the focal depth at surface S A , on which thevirtual receivers are located. These receivers are usedto measure the up- and down-going Green’s functions: G − ( x A , x R , t ) and G + ( x A , x R , t ), respectively. Here, x A is the location of the virtual receivers at the focaldepth. For the definition of the focusing functions, themedium is truncated below the focal depth, resultingin a medium that is inhomogeneous between S and S A , and homogeneous above and below these surfaces.In this medium we define a downgoing focusing func-tion f +1 ( x R , x A , t ), which, when injected from the sur-face, focuses at the focal depth S A at x A . Moreover, f − ( x R , x A , t ) is the upgoing response of the mediumas measured at the surface, known as the upgoing fo-cusing function. These ideas can be combined in twointegral equations, as follows (Wapenaar et al., 2014;Slob et al., 2014): G − ( x A , x R , t ) + f − ( x R , x A , t ) = Z S R ( x R , x S , t ) ∗ f +1 ( x S , x A , t ) d x S , (1) G + ( x A , x R , t ) − f +1 ( x R , x A , − t ) = − Z S R ( x R , x S , t ) ∗ f − ( x S , x A , − t ) d x S . (2)For acoustic media, the focusing- and Green’s func-tions on the left-hand side are separable in time by awindowing function. In practice, the infinite integralson the right-hand side are approximated by a finitesum over the available sources: X i R ( x R , x ( i ) S , t ) ∗ f +1 ( x ( i ) S , x A , t ) ∗ S ( t ) , (3) − X i R ( x R , x ( i ) S , t ) ∗ f − ( x ( i ) S , x A , − t ) ∗ S ( t ) , (4) where i denotes the source position and S ( t ) the sourcesignature. When the reflection response is not wellsampled, these summations cause distortions in theresponses on the left-hand sides of Equation 1 and 2. Point-spread functions
Wapenaar and van IJsseldijk (2020) introduce point-spread functions (PSFs) to correct for imperfect sam-pling. These PSFs exploit the fact that the downgo-ing focusing function is the inverse of the transmis-sion response. A convolution of the focusing functionwith the transmission response should, therefore, givea delta pulse in space en time. However, for imper-fectly sampled data this delta pulse gets blurred. Thisblurring describes the imperfect sampling, as follows:Γ + ( x A , x A , t ) = X i T ( x A , x ( i ) S , t ) ∗ f +1 ( x ( i ) S , x A , t ) ∗ S ( t ) . (5)Here Γ + and T are the downgoing PSF and trans-mission response, respectively. Similarly, a quantity Y is defined as the inverse of the time-reversed, up-going focusing function. The convolution to quantifythe upgoing PSF (Γ − ) then becomes:Γ − ( x A , x A , t ) = X i Y ( x A , x ( i ) S , t ) ∗ f − ( x ( i ) S , x A , − t ) ∗ S ( t ) . (6)Once again, in the case of perfect sampling this PSFwould be equal to a delta pulse in space and time.Note that this inverse ( Y ) is not necessarily stable,because f − is a reflection response. On the contrary, f +1 is more stable and better invertible (in the limitingcase of a 1D medium it is a minimum-phase function,which is always invertible). This will be elaboratedupon in the discussion section.Next, Wapenaar and van IJsseldijk (2020) apply thesenewly acquired PSFs to Equation 1 and 2, respectively.This results in two new representations: “ G − ( x A , x R , t ) + “ f − ( x R , x A , t ) = X i R ( x R , x ( i ) S , t ) ∗ f +1 ( x ( i ) S , x A , t ) ∗ S ( t ) , (7) “ G + ( x A , x R , t ) − “ f +1 ( x R , x A , − t ) = − X i R ( x R , x ( i ) S , t ) ∗ f − ( x ( i ) S , x A , − t ) ∗ S ( t ) , (8)with: “ G ± ( x A , x R , t ) = Z S A G ± ( x A , x R , t ) ∗ Γ ∓ ( x A , x A , t ) d x A , (9) “ f ± ( x R , x A , ∓ t ) = Z S A f ± ( x R , x A , ∓ t ) ∗ Γ ∓ ( x A , x A , t ) d x A . (10) 𝒇 = 𝜽 𝑮 − + 𝒇
𝟏− 𝒌 𝒇 = 𝑮 𝒅 −𝟏 ҭ𝑮 − + ҭ𝒇
𝟏− 𝒌 = 𝑹 ∗ 𝒇 ∗ 𝑺𝑘 = odd 𝑘 = even 𝑻 𝒌 = 𝒇 𝒀 𝒌 = 𝒇 − ★ −𝟏 𝜞 𝒌+ = 𝑻 𝒌 ∗ 𝒇 𝜞 𝒌− = 𝒀 𝒌 ∗ 𝒇 ★ 𝑮 − + 𝒇
𝟏− 𝒌 = ҭ𝑮 − + ҭ𝒇
𝟏− 𝒌 𝜞 𝒌+ − ҭ𝑮 + + ҭ𝒇 ★ 𝒌 =− 𝑹 ∗ 𝒇 ★ ∗ 𝑺𝒇 = 𝒇 +𝜽 −𝑮 + + 𝒇 ★ 𝒌 ★ −𝑮 + + 𝒇 ★ 𝒌 = − ҭ𝑮 + + ҭ𝒇 ★ 𝒌 𝜞 𝒌 − Figure 1: Flowchart with the proposed iterativeMarchenko scheme, step 3 to 5 account for imperfectlysampled data. f and G represent the focusing- andGreen’s functions, respectively. S is the source signa-ture. k denotes the iteration number. The arch over aletter denotes that the response is contaminated by theimperfect sampling, the superscript star denotes time-reversal. The asterisks denote convolutions or corre-lations, which are then summed over the imperfectlysampled sources. θ is the time-windowing operator.Equations 7 and 8 have two interesting features. First,the right-hand sides are now the same as Equations3 and 4. Second, the responses on the left-hand sidesnow contain the PSFs, which apply a blurring effect toeach response. Note that the imperfectly sampled datacan now be deblurred by a multidimensional deconvo-lution (MDD) with the PSFs, assuming these PSFs areknown. Iterative Marchenko scheme
Wapenaar and van IJsseldijk (2020) verify the rep-resentations in equations 7 and 8, using analyticallymodelled focusing functions (i.e. both the reflectionresponse and focusing functions on the RHS of theequations are known). In practice, these focusing func-tions are unknown, and have to be retrieved from theMarchenko equations. This can be achieved iterativelyor by inversion of the Marchenko equations. Here, weaim to integrate the representations for imperfectlysampled date with the iterative approach (Thorbeckeet al., 2017).Figure 1 shows the proposed iterative Marchenko scheme,which corrects for imperfect sampling in each iteration -3000 -2000 -1000 0 1000 2000 3000Lateral distance [m] -3000 -2000 -1000 0 1000 2000 3000Lateral distance [m]02004006008001000120014001600 D e p t h [ m ] ] Figure 2: Model used in the irregular sampling exper-iment, the dashed red line shows the focal level. Thebarcode shows the irregular sampling, with the whitespaces denoting the excluded sources. k . The first step is to estimate the initial downgo-ing focusing function ( f +1 , ). Traditionally, this is esti-mated by the time-reversal of the direct arrival of theGreen’s function. However, to ensure that the con-volution of the transmission response and downgoingfocusing function gives a delta pulse in space and timewith the correct amplitudes, the proposed scheme in-verts the direct arrival in step 1: f +1 , ( x R , x A , t ) ≈ { G d ( x A , x R , t ) } − . (11)The next step computes the focusing- and Green’sfunction by a convolution or correlation for the odd oreven iterations, respectively. The odd iterations arecomputed according to Equation 7, where the down-going focusing function on the RHS is retrieved fromthe initial condition for the first iteration or from theprevious iteration for subsequent iterations. Similarly,the even iterations use the upgoing focusing functionsfrom the previous iteration in the correlation with thereflection response, as shown in Equation 8. Note, forwell-sampled data the computed focusing- and Green’sfunctions in this step are free of distortions, thereforethe resulting focusing- and Green’s functions are equalto these functions in the standard scheme: { “ G ± ( x A , x R , t ) ∓ “ f ± ( x R , x A , ∓ t ) } k = { G ± ( x A , x R , t ) ∓ f ± ( x R , x A , ∓ t ) } k . (12)In this case steps 3 to 5 are redundant and can beomitted, this indeed reduces the proposed scheme tothe standard iterative Marchenko scheme.For irregular sampled reflection data steps 3 to 5 areintroduced. The first objective is to find an estimate ofthe transmission response and quantity Y for odd andeven iterations, respectively. Since these responses aredefined as the inverse of the focusing functions, theycan be obtained by inversion of the following equa-tions: δ ( x H,A − x H,A ) δ ( t ) = Z S T k ( x A , x S , t ) ∗ f +1 ,k − ( x S , x A , t ) d x S , (13)and δ ( x H,A − x H,A ) δ ( t ) = Z S Y k ( x A , x S , t ) ∗ f − ,k − ( x S , x A , − t ) d x S . (14) T k in Equation 13 denotes the estimated transmissionresponse for each odd iteration k , and f +1 ,k − is thedowngoing focusing function computed in the formeriteration k −
1. Equation 14 computes an approxi-mate of the quantity Y k for each even iteration, basedon the upgoing focusing function from the precedingiteration. Note that both the up- and downgoing fo-cusing functions are deblurred, and free of distortionsfrom the imperfect sampling. The two integrals repre-sentations are, therefore, evaluated over a regular grid(e.g. as if no sources are missing). Next, the PSFshave to be computed, using the estimates of T and Y (step 4 in Figure 1. Analogous to Equation 5, thedowngoing PSF for each odd iteration is retrieved byevaluation the convolution of T k and f +1 ,k − over theirregular sampled sources. For the even iterations weconsider the correlation of Y k and f − ,k − , as in Equa-tion 6. Subsequently, in step 5 the distorted focusing-and Green’s functions, from step 2 of the scheme, aredeblurred by a multidimensional deconvoltion with thePSFs. Thus, the resulting focusing- and Green’s func-tions are reconstructed as if they were retrieved withwell-sampled data. Finally, the last step separates thefocusing function from the Green’s function using atime-windowing operator ( θ in Figure 1). This finalstep is identical to that in the standard Marchenkoscheme.Each iteration is initialized with a "clean" focusingfunction from the preceding iteration. This is requiredat the start of each iteration, otherwise the errors fromthe irregular sampled reflection data would accumu-late. Therefore, steps 3 to 5 are enforced with everyiteration, as opposed to only a single time after alliterations are finished. Numerical example
The performance of the proposed scheme is tested onsynthetic data, applying the new methodology pro-posed in this work. The 2D model for this test isshown in Figure 2. For convenience, the density andvelocity parameters are chosen to be the same in each layer, but this is not required for successful applica-tion of the scheme. The observant reader will note thestrong contrast in acoustic impedance between the toptwo layers of the model, at a depth of 200 meter. Thiscontrast ensures that the inversion of f − for retrieving Y is stable, because most of the energy gets concen-trated at the early onsets of the reflection response.The reflection response of the medium is modeled us-ing a wavelet with a flat spectrum between 5 and 80Hz, after which the direct wave is removed. In total601 sources and receivers are used with an initial spac-ing of 10 meters. For the irregular sampling 50% ofthe sources are removed at random, as can be seen inthe barcode plot in Figure 2. Next, the direct arrivalof the Green’s function between the focal depth andthe Earth’s surface is estimated in a smooth velocitymodel. As previously stated, the inverse of this directarrival is used for the initial estimate of the upgoingfocusing function, as opposed to the time-reversed ver-sion that is traditionally used. The reflection responseand this initial estimate together are all the requiredinputs for the standard Marchenko scheme. Finally,for the fourth step of our proposed scheme the loca-tion of the sources (e.g. the barcode in Figure 2) isrequired. Results
Figure 3 shows the results of the numerical experi-ment, each column in the figure represents the re-sults after 12 iterations using one of the three schemes.The first column shows the results where the standardMarchenko scheme is used with the irregularly sam-pled reflection data. Next, the middle column showsthe results of the proposed scheme, again with irregu-larly sampled data. Finally, the last column displays areference result, that was obtained by using the stan-dard scheme on reflection data without removing anysources. The red dashed line in the figure denotesthe seperation in time of the Green’s functions below,and focusing functions above. In the case of irregu-lar sampling in the standard scheme (as presented inthe first column), three main artifacts can be identi-fied. Firstly, clear distortions of some reflectors areobserved, especially around the strong events. Thesedistortions are most noticeable of all artifacts, and ob-struct later events in the downgoing Green’s function( “ G + ). In Figure 3 the ellipses indicate some of theseartifacts. Secondly, the amplitudes of some events areincorrect or the events are not reconstructed at all (asshown by the red arrows in the figure). For example,the downgoing focusing function ( “ f + ) is largely sup-pressed, as well as some events in the upgoing focusingfunction ( “ f − ). Lastly, some new and undesired reflec-tors are appearing in the results, especially at latertimes ( > . “ G − ) are deviating from the refer-ence result in the third column. Examples of such un- -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f +1 G + -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f +1 G + -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f +1 G + -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f +1 G + -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f +1 G + -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f +1 G + -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f +1 G + -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f +1 G + -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f G -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f G -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f G -2000 -1000 0 1000 2000Lateral distance [m]-0.80.00.81.62.4 T i m e [ s ] f G Figure 3: The top row shows the time-reversed downgoing focusing function ( { f +1 } ? ) and downgoing Green’sfunction ( G + ), and the bottom row shows the upgoing focusing function ( f − ) and upgoing Green’s function ( G − ),the star superscript denotes time-reversal. The dashed, red lines indicate the separation between the focusing-and Green’s functions. The left columns show the result of irregularly sampled data after 12 iterations of thestandard Marchenko scheme. The middle columns show the results when using our scheme on the same data(Figure 1), again 12 iterations are used. Finally, the 3rd column shows the reference result, obtained after 12iterations of the standard Marchenko scheme with well-sampled data. Each panel is scaled with it’s maximumvalue. The arrows and ellipses show artifacts arising from the irregular sampling. Distortions caused by theirregular sampling are indicated with the ellipses. The red arrows show events that deviate in amplitude or aremissing altogether. Finally, the blue arrows mark erroneous reflectors. Amplitude0.250.000.250.500.751.001.25 T i m e [ s ] Regular Marchenko Irregular with PSFs Irregular no PSFsAmplitude0.250.000.250.500.751.001.25 T i m e [ s ] Amplitude0.250.000.250.500.751.001.25 T i m e [ s ] Regular Marchenko Irregular with PSFs Irregular no PSFsAmplitude0.250.000.250.500.751.001.25 T i m e [ s ] Amplitude0.250.000.250.500.751.001.25 T i m e [ s ] Regular Marchenko Irregular with PSFs Irregular no PSFsAmplitude0.250.000.250.500.751.001.25 T i m e [ s ] Amplitude0.250.000.250.500.751.001.25 T i m e [ s ] Regular Marchenko Irregular with PSFs Irregular no PSFsAmplitude0.250.000.250.500.751.001.25 T i m e [ s ] Figure 4: Comparison of the amplitudes in the middletrace (at offset 0 m) of each panel in Figure 3. On theleft are the time-reversed downgoing focusing function( { f +1 } ∗ ) and downgoing Green’s function ( G + ). Theupgoing focusing function ( f − ) and upgoing Green’sfunction ( G − ) are shown on the right.desired reflectors are marked with the blue arrows. Allthree types of these artifacts are mostly removed byusing the proposed scheme (middle column), and theresults of this scheme show much more resemblancewith the reference results. This implies that the pro-posed scheme both deblurs the results of irregular sam-pling effects, and also retrieves the amplitudes of theevents more accurately. However, the method doesintroduce some of it’s own artifacts; as it introducesedge effects, especially at later times. These artifactsare introduced by the MDD of poorly sampled datawith the PSFs, and they are suppressed by using di-rectional FK-filters.The amplitude reconstruction by the proposed schemeis further illustrated in Figure 4, where the middletrace of each panel from Figure 3 is plotted. In Fig-ure 4 the results of the proposed scheme in orangequite closely match the reference results in blue. Whereasthe standard scheme fails to recover the correct ampli-tudes in the case of irregularly sampled reflection data(green line). This difference in amplitudes cannot sim-ply be negated by scaling with a constant factor, be-cause the error has a different magnitude at differenttimes. Discussion
The results show that the proposed scheme can suc-cessfully be used on irregularly sampled reflection data.However, the new method has some limitations, andthere are possibly some improvements that allow forbetter results.First, the largest limitation of our method is the in-stability of quantity Y , which was introduced as theinverse of the upgoing focusing function. The densi-ties and velocities of the test model were restrainedto ensure stability of Y . A possible solution to thisproblem is to include free-surface multiples within theiterative scheme (Singh et al., 2015), which would nolonger require the inversion of f − . It is noted that thisinclusion can lead to instabilities in the Marchenko se-ries (e.g. Staring et al., 2017; Dukalski and de Vos,2017), but it is expected to be more stable than thecurrent inversion. However, this is subject to ongoingresearch.While the inverse of the downgoing focusing functionalways exists, there is a different way to estimate thetransmission response, which does not require any ex-plicit inversions (Vasconcelos et al., 2018). This method-ology was also tested to calculate the transmission re-sponse in step 3 of the proposed scheme. While thismethod achieved promising results in 1.5D media, wefound that the results were unsatisfactory in the 2Dmodel. Therefore, the transmission response was esti-mated by inversion instead.The new methodology is unable to account for irreg-ular sampling of both sources and receivers; the sam-pling can only be irregular in the same dimension asthe integration in equations 1 and 2. On the con-trary, the method introduced by Haindl et al. (2018)requires irregular sampling in the opposite dimension.A combination of these complementary methods is,therefore, envisioned to deal with irregular samplingin both the source and receiver dimensions simulta-neously. However, further research into this topic isrequired.Finally, we note that the reflection data can also bereconstructed before applying the Marchenko method.This interpolated reflection response could then beused in the standard iterative scheme, but would re-quire additional pre-processing. This approach hasbeen tested by Haindl (2016), who found that the re-sulting Green’s and focusing functions contained a rel-atively high level of noise. Conclusion
One of the restrictions of the Marchenko method is theneed for well-sampled and collocated sources and re-ceivers. Recent work introduced new representationsfor irregularly sampled data. These representations in-cluded point-spread functions (PSFs) that deblur dis-torted focusing- and Green’s functions. Based on theserepresentations, this paper showed that the iterativeMarchenko scheme can be adapted to handle irregu-larly sampled data. This adaptation introduces a fewadditional constrains to the Marchenko method: Thelocation of the missing sources needs to be known, andan inverse version as opposed to the time-reversed ver-sion of the direct arrival of the Green’s function is re-quired as initial estimate of the scheme. In addition,each iteration of the scheme is extended by three steps.First, an approximation of the transmission responseor quantity Y needs to be computed for the odd andeven iterations, respectively. Quantity Y is the inverseof the upgoing focusing function, similar as the trans-mission response is the inverse of the downgoing focus-ing function. Second, these approximations are irregu-larized in accordance with the missing sources. Subse-quently, these irregular versions are used to calculatea PSF. Third, the well-sampled focusing- and Green’sfunctions are reconstructed by a multidimensional de-convolution of the blurred these functions with thePSFs.The newly proposed scheme alleviates the need forwell-sampled sources when using the Marchenko method.Ideally, the need for well-sampled receivers should beremoved as well. While this is subject to ongoing re-search, a new scheme involving a sparse inversion isenvisioned. By relaxing the need for perfectly sampleddata, the Marchenko method is more easily applied tofield data. Acknowledgements
The authors thank Jan Thorbecke and Christian Reinickefor help with the numerical examples and insightfuldiscussions. This research was funded by the Euro-pean Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation pro-gramme (grant agreement No: 742703).
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