on sampling functions and fourier reconstruction...

On sampling functions and Fourier reconstruction methods

Mostafa Naghizadeh ∗ and Mauricio D. Sacchi ∗

ABSTRACT

Random sampling can lead to algorithms where the Fourier reconstruction is almostperfect when the underlying spectrum of the signal is sparse or band-limited. Con-versely, regular sampling often hampers the Fourier data recovery methods. However,two-dimensional (2D) signals which are band-limited in one spatial dimension can berecovered by designing a regular acquisition grid that minimizes the mixing betweenthe unknown spectrum of the well-sampled signal and aliasing artifacts. This conceptcan be easily extended to higher dimensions and used to define potential strategiesfor acquisition-guided Fourier reconstruction. In this paper we derive the wavenumberresponse of various sampling operators and investigate sampling conditions for optimalFourier reconstruction using synthetic and real data examples.

INTRODUCTION

The sampling theorem is an interesting topic of chief importance in the physical sciences andengineering. Sampling methods can be classified into two categories, uniform and nonuni-form sampling methods. In uniform sampling methods, a signal is sampled periodically at aconstant rate. This leads to recovery conditions based on the well-known Nyquist samplingtheorem. An overview of uniform sampling and its properties is given by Unser (2000). Fornonuniform sampling methods, a distinction is made between methods that assume knownsampling positions, and those where the position at which observations were taken are notprecisely known (Vandewalle et al., 2007). In this paper, we will concentrate on methodswith known sampling locations.

Multi-channel sampling is another form of sampling that leads to the so called super-resolution reconstruction techniques (Bertero and Boccacci, 2003; Park et al., 2003). Ageophysical example of multi-channel sampling was provided by Ronen (1987) who proposedan algorithm to estimate an alias-free zero-offset section from aliased common offset seismicsections. This algorithm was later developed into inversion to zero-offset (Ronen et al.,1991, 1995).

Data reconstruction methods based on signal processing concepts have been proposedto reconstruct seismic data. Two important categories are prediction filtering methods,and transform-based methods. Prediction filtering approaches have been proposed by Spitz(1991); Porsani (1999). The frequency-wavenumber (f-k) equivalence of prediction filteringmethods is introduced by Gulunay (2003). Their techniques are capable of interpolatingregularly sampled aliased data. These type of methods have been modified by Naghizadehand Sacchi (2007) to cope with irregularly sampled data. Transform-based methods mapthe signal to a new domain and a synthesis of the signal from the transformed domainis used to generate data at un-recorded spatial locations. Transform-based seismic data

Fourier reconstruction

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reconstruction have been proposed by, to cite a few, Bardan (1987), Darche (1990), andTrad (2008). Transform-based methods which operate with Fourier basis have been alsoadapted for seismic data reconstruction. In this case the seismic signal is represented via asuperposition of complex exponentials.

Fourier reconstruction methods rely on two basic assumptions. In general, we assumeband-limited signals in the wavenumber domain and/or signals that can be represented by aparsimonious distribution of Fourier coefficients. Examples of the aforementioned assump-tions abound in the signal and imaging processing literature. For instance, reconstructionof band-limited signals via Fourier methods have been studied by Strohmer (1997); Feuerand Goodwin (2005); Eldar (2006). In the geophysical literature Duijndam et al. (1999) andSchonewille et al. (2003) proposed a band-limited signal reconstruction method for seismicdata that depends on two and three spatial dimensions. Methods that used the sparsityassumption to reconstruct data were proposed by Sacchi and Ulrych (1996) and Sacchiet al. (1998) and expanded to the multidimensional case by Zwartjes and Gisolf (2006) andZwartjes and Sacchi (2007).

The intention of this paper is to provide an understanding of the importance of sam-pling at the time of defining a Fourier reconstruction strategy. In particular we investigatedifferent type of samplings and the impact they have on the accuracy of the general form ofband-limited Fourier reconstruction introduced by Liu (2004). The latter, also called Min-imum Weighted Norm Interpolation (MWNI) method, permits to retrieve the wavenumberspectrum of desired data from a decimated data. It must be mentioned that this articleonly investigates the MWNI method. The conclusions arising from our analysis, however,are valid for other reconstruction methods that adopt a simple or sparse spectral represen-tation. This includes the anti-leakage Fourier transform (Xu et al., 2005), projection ontoconvex sets (Abma and Kabir, 2006) and other sparsity promoting solutions to the interpo-lation problem (Sacchi et al., 1998; Hennenfent, 2008). Also, our general conclusions mayapply to methods that use other transforms such as wavelets (Beylkin et al., 1991), curvelets(Candes, 2006), seislets (Fomel, 2006) and etc. Reconstruction methods that adopt sparsitypromoting strategies are part of the general theory of compressive sensing (Donoho, 2006;Tsaig and Donoho, 2006). The latter deals with the optimal recovery conditions for signalsthat admit sparse representation in surrogate domain.

In our paper, random sampling means randomly picked samples from a regular grid ofdata. A regular grid of data is a requirement for the MWNI method (Liu, 2004). The latterpermits to design an computationally efficient interpolator that heavily relies on spatialfast Fourier transforms (FFT). The anti-leakage Fourier transform reconstruction (Xu et al.,2005), on the other hand, uses the exact irregular spatial positions and consequently requiresoperators based on non-uniform discrete Fourier transforms. It must be mentioned that inreal practical surveys the pure irregularity of spatial sampling might completely eliminatethe alias problem. However, in most of seismic surveys, the spatial sampling operators arenear regular and therefore they might suffer from the same problems of regular grids.

This paper is organized as follows. First, we review one-dimensional and two-dimensionalsampling operators and their frequency or wavenumber responses. We examine the wavenum-ber domain footprint of the sampling operators and study cases where MWNI can properlyreconstruct data from under-sampled observations. We then provide examples of syntheticand real data, highlighting our finding on the relationship between sampling operators andFourier reconstruction.

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SAMPLING THEORY

One-dimensional sampling functions

We start our discussion by considering a discrete signal x of length N

x = {x(0), x(1), x(2), . . . , x(N − 1)}. (1)

In addition, we will replace N − M arbitrary samples of x by zeros to simulate miss-ing observations. The location of the available samples are indicated by the set H ={h(0), h(1), h(2), . . . , h(M − 1)}. The signal with missing samples is represented via thefollowing expression

xs(n) ={

x(h(n)) h(n) ∈ H0 h(n) /∈ H . (2)

The discrete Fourier transform (DFT) of x is defined by

X(k) =1N

N−1∑n=0

x(n)e−i2πnk

N k = 0, 1, 2, . . . , N − 1. (3)

We now we compute the DFT of signal with missing samples

Xs(k) =1N

M−1∑n=0

x(h(n))e−i2πh(n)k

N , k = 0, 1, 2, . . . , N − 1, (4)

It is easy to show (Appendix) that the DFT of the sampled signal can be representedin vector form as follows

Xs = X ∗Q, (5)

where the symbol ∗ indicates discrete periodic convolution (Oppenheim and Schafer, 1974)and Q denotes the Fourier response of sampling function with elements given by

Q(k) =1N

M−1∑m=0

e−i2πh(m)k

N k = 0, 1, 2, . . . , N − 1. (6)

The sampling function in equation 6 convolves the spectrum of the original signal to yield thespectrum of the signal with missing samples. This equation is fundamental to understandingFourier reconstruction methods. Let’s use examples to study the footprint of the samplingoperator in the wavenumber domain. Figure 1 shows sampling functions (left side) and theircorrespondent frequency responses (right side). In this example we assume that the data hasbeen regularly decimated. The decimation factors are 1, 2, 3, 4 and 5 from top to bottom,respectively. For the decimation factor equal to 1 (no decimation), the wavenumber responseof the sampling function has only one impulse at the normalized wavenumber k = 0.0. Thismeans that the DFT vectors Xs and X are equal. When the data are decimated by a factorof 2, every other sample is replaces by zeros and the wavenumber response contains twoimpulses. In other words, the DFT of the sampled data will contain replicas of the DFT of

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Figure 1: Sampling operators (left) and their associated wavenumber spectra (right). The decima-tion factors are 1, 2, 3, 4 and 5 from top to bottom.

the data prior to decimation. Increasing the decimation factor results in more replicationsof the original spectrum. It is important to notice that the size of replicated impulses ispreserved. If KN denotes the Nyquist wavenumber, the decimation process by an integerfactor L of a periodic sequence can also be described as the lateral cyclic rotation of theoriginal wavenumber spectrum by amounts KN/L, 2KN/L, . . . , (L − 1)KN/L followed bysummation (Gulunay, 2003).

Figure 2 depicts 5 different random sampling scenarios (left side) and their wavenumberresponses (right side). For all of the examples, 50% of the samples are missing. Thewavenumber response of a nonuniform sampling operator has a different structure than theuniform one. It has nonzero values for the all of its components inside the fundamentalinterval of the wavenumber spectrum. Interestingly, it has only one large value whichappears at the exact location k = 0. The rest of the components are small. As the numberof missing samples increases, more components of the sampling function are dominant.Figure 3 shows 5 random sampling operators with 85% missing samples. As the numberof missing samples increases, the amplitude of the impulses in the wavenumber response ofthe random sampling operator increases.

To continue with the analysis we now examine the case where the data contain gaps. Fig-ure 4 shows the plot of various sampling functions (left side) and their associated wavenum-ber responses (right side). Similarly, Figure 5 depicts the plot of different sampling functions(left side) for the extrapolation case and their corespondent wavenumber responses (rightside). Both cases produce wavenumber responses with a dominant amplitude responsesat the negative and positive normalized wavenumbers near k = 0. The spectrum of thedata that has undergone this type of sampling will be blurred by the convolution with thewavenumber response of the sampling function. On the other hand, in the regularly sam-

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Figure 2: Sampling operators (left) and their associated wavenumber spectra (right). For eachexample 50% of data were randomly removed.

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Figure 3: Sampling operators (left) and their associated wavenumber spectra (right). For eachexample 85% of data were randomly eliminated.

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Figure 4: Sampling operators (left) and their associated wavenumber spectra (right) for a gap insidethe data. The size of the gap decreases from top to bottom.

pled case the spectrum of the sampled data will exhibit, depending on wavenumber support,organized replicas (alias), which can only be removed when there is no overlap between thespectrum of the signal and the aliased components (band-limited signal). In other words,in order to interpolate the data, such overlaps must be suppressed. (Gulunay, 2003).

Multidimensional sampling functions

The formulas introduced in the previous section can be easily generalized to multidimen-sional cases. We will concentrate on the 2D case but bear in mind that a generalization toN-D case is straightforward. Let us consider the lattice u, with size Nx × Ny representedby

u(nx, ny) ={

nx ∈ 0 : Nx − 1ny ∈ 0 : Ny − 1

. (7)

Assume that we arbitrarily reserve M samples of data and set the rest to zero to obtain us,the signal with missing samples. Let us indicate the location of the available samples by

H = {(hx(m), hy(m)) | m ∈ 0 : M − 1}, (8)

where, hx and hy indicate the location of the available samples on the axis X and Y ,respectively. The DFT of the original data, U, is given by:

U(kx, ky) =1

Ny

Ny−1∑ny=0

(1

Nx

Nx−1∑nx=0

u(nx, ny)e−i2π kxnx

Nx

)e−i2π

kynyNy . (9)

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Figure 5: Sampling operators (left) and their associated wavenumber spectra (right) for the ex-trapolation case. The number of missing samples is decreasing from top to bottom.

Now, the DFT of the data with missing samples, Us, is obtained by

Us = U ~ Q, (10)

where, ~ represents the 2D convolution operator and Q is the 2D Fourier response givenby

Q(kx, ky) =1

NxNy

M−1∑m=0

e−i2πkxhx(m)

Nx e−i2π

kyhy(m)

Ny . (11)

In this paper the absolute value (or amplitude) of the wavenumber response of the 2Dsampling function, Q, is plotted for various sampling scenarios. To start the analysis, Figure6b shows the wavenumber response of a sampling function with no missing samples (Figure6a). The wavenumber response is a single impulse in the middle of the 2D spectrum.Figure 6c shows a 2D sampling operator that eliminates every other slice of data in theY direction. Filled circles represent the available samples and crosses show the missingsamples. Figure 6d shows the wavenumber response of Figure 6c. The wavenumber responsehas two impulses located at normalized wavenumbers (kx, ky) = {(0, 0), (0,−0.5)}. Figure6e shows another 2D decimation function that samples the data in a chessboard pattern.Figure 6f shows the wavenumber response of Figure 6e. In comparison to Figure 6d, theimpulse at the normalized wavenumber (0,−0.5) is moved to (−0.5,−0.5).

This is an interesting phenomenon and can be efficiently utilized for data reconstructionpurposes. Assume that the original 2D signal in Figure 6b is band-limited in the kx axis inthe interval [−0.2, 0.2] and full band in the ky axis. Therefore, in Figure 6d the replicationimpulse produce by the sampling operator is inside the interval where the signal lives. InFigure 6f, however, the impulse resides outside the region containing the signal. Hence,

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in the case of Figure 6f one use a band-limiting constraint on the kx axis to remove thesampling artifacts. Conversely, in Figure 6d band-limitation cannot be used to eliminatethe sampling artifacts.

Figure 7a shows a sampling function which has eliminated two Y slices between eachavailable Y slice. Figure 7c shows another sampling function which also has eliminated twosamples between available samples and the available samples are located in all of the Yand X slices. Figures 7b and 7d show the wavenumber responses of the Figures 7a and 7c.In Figure 7d the band-limitation information on the kx axis can be used to eliminate theartifacts. This is not possible in the situation depicted in Figure 7b.

To continue with our analysis, we now examine the case where samples from the gridare randomly eliminated. Figures 8a and 8c show two 2D random sampling operators with50% and 85% missing samples, respectively. Figures 8b and 8d depict the wavenumberresponses of 8a and 8c, respectively. It is interesting that in both wavenumber responsesthe amplitude of impulses are very small compared to the impulse of the original datalocated in the center of the plots. Therefore, for randomly sampled multidimensional datawith a high percentage of missing samples, the spectral distortion could be minimal. Figure8e shows a 2D sampling operator for a gap inside the data. The wavenumber response of 2Dgap sampling (Figure 8f) contains relatively high amplitude impulses only in the vicinity ofthe main original impulse.

Band-limited Minimum Weighted Norm Interpolation

In the previous section we have reviewed basic concepts in sampling theory. In this sec-tion we examine how one can reconstruct seismic data using MWNI for different samplingscenarios. In particular, we will highlight the importance of understanding the interplaybetween sampling and a given reconstruction algorithm for a successful reconstruction ofseismic data. The MWNI method is described in detail in Liu (2004), Liu and Sacchi (2004)and Trad (2009). Interpolation of band-limited data with missing samples using the MWNImethod is summarized in the following paragraphs.

The observed data dobs is related to the unknown data in the desired grid, d, via thesampling operator G. The desired data can be retrieved by minimizing the following costfunction

J = ||Gd− dobs||22 + µ ‖d‖2W , (12)

where ‖.‖2W indicates a weighted norm and G is the sampling matrix which maps desired

data samples d = d(xn, f) to available samples dobs = d(xh, f) at a given frequency f . Itstranspose, GT , fills the position of missing samples with zeros (Liu and Sacchi, 2004). Thenorm is expressed as follows

‖d‖2W =

∑k∈K

D∗kDk

W 2k

, (13)

where Dk indicates the coefficients of the Fourier transform of the vector of spatial data d.

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Figure 6: a) Full 2D sampling operator. c) 2D regular sampling operator with a decimation offactor 2 in the Y direction. e) 2D regular sampling operator with chessboard pattern decimation.b), d), and f) are 2D wavenumber spectra of (a), (c), and (e), respectively.

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Figure 7: a) and c) Two different 2D regular sampling functions with a decimation factor of 3. b)and d) are the wavenumber spectra of (a) and (c), respectively.

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Figure 8: a) and c) Two 2D random sampling operators with 50% and 85% of missing samples,respectively. e) 2D gap sampling operator. b), d), and f) are 2D wavenumber spectra of (a), (c),and (e), respectively. The second largest amplitudes in (b) and (d) are 0.15 and 0.33, respectively.

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For the MWNI method the diagonal matrix is defined as

Υk ={

W 2k k ∈ K

0 k /∈ K , (14)

where K indicates the region of support of the Fourier transform. The pseudoinverse of Υis defined as:

Υ†k =

{W−2

k k ∈ K,0 k /∈ K.

(15)

For the band-limited MWNI, the values of Wk are equal to one, while for the MWNI method,their values must be iteratively updated to find an optimal reconstruction. The minimizerof the cost function in expression 12 is given by

d̂ = FHΥFGT (GFHΥFGT + αI)−1dobs, (16)

where F is the Fourier transform operator in matrix form, α is trade-off parameter, Iis the identity matrix, T and H stand for transpose and Hermitian transpose operators,respectively. For further details see Liu (2004) and Liu and Sacchi (2004). Tables 1 -3 demonstrate the Conjugate Gradients (CG) algorithm used to implement the MWNImethod. The symbols ~ and � represent the circular convolution and element by elementmultiplication, respectively.

EXAMPLES

Synthetic 1D examples

Random sampling operator

In order to examine the performance of the MWNI method with various sampling opera-tors, a synthetic and real-valued 1D signal with two harmonics was created (Figure 9a). Thedata contains 120 samples and the harmonics are at 0.06 and 0.13 normalized wavenumberswith amplitudes equal to one. For our 1D examples, the left panels will show the data inthe spatial domain (where we have assumed ∆x = 1) and the right panels will representtheir correspondent normalized wavenumber domains. To proceed with the first example,80 percent of the data was eliminated randomly. The missing samples (Figure 9c) werereconstructed using the MWNI method. Figure 9e shows the result of the reconstruction.Since available samples were picked by a random nonuniform sampling operator the recon-struction was successful. This is due to the fact that the wavenumber response for thenonuniform sampling operator has only one high amplitude impulse (Figure 2). Hence, itsconvolution with the spectrum of the original data (which is simple as well) will create asimple signal (with impulses at the same location of the impulses of the original spectrum)suitable to be reconstructed using the MWNI method. Figures 9b, 9d, and 9f show theFourier panels of Figures 9a, 9c, and 9e, respectively.

There is a common misinterpretation pertaining random sampling, which needs to beclarified. One might think that the shortest distance between consecutive samples (or av-erage sampling rate) determines the success of a reconstruction method based on sparsityor MWNI. Figure 10a shows an example of randomly picked samples in which the shortest

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Table 1: Conjugate Gradients algorithm for the MWNI method

Input:Total number of data samples: ntIndices of the available samples: hns×1

Values of the available samples: yns×1

Hanning window: Hnh×1

Number of external and internal iterations: niter1, niter2

Error reduction toleration: tol1, tol2Output:Reconstructed data: x̂begin

Initiate band-limiting function : Υnt×1 ={

1 Desired wavenumbers0 Elsewhere

for i = 1 to niter1

X = 0nt×1 (Initiation with zero vector)s = yR = Forward Operator(s,h,Υ, nt) (Table 2)P = Rε1 = R ·R∗

δ1 = s · s∗for j = 1 to niter2

q = Adjoint Operator(P,h,Υ) (Table 3)α = ε1/(q · q∗)X = X + αPs = s− αqR = Forward Operator(s,h,Υ, nt) (Table 2)ε2 = ε1ε1 = R ·R∗

β = ε1/ε2P = R + βPδ2 = s · s∗if [(δ1 − δ2) < tol1] or [δ2 < tol2] then

Breakendδ1 = δ2

endUpdate and smooth band-limiting function:Υ = abs(X) ~ H

endx̂ = IFFT (X)

end

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Table 2: Forward operator of CG algorithm for the MWNI method.

begint = 0nt×1

t(h) = sT = FFT (t)R = Υ�T

end

Table 3: Adjoint operator of CG algorithm for the MWNI method.

beginT = Υ�Pt = IFFT (T)qns×1 = t(h)

end

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Figure 9: a) Original data. c) Data with 80% randomly missing samples. e) Reconstructed datavia MWNI. b), d), and f) are the wavenumber spectra of (a), (c), and (e), respectively.

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distance between consecutive samples was forced to be more than 4 samples. The recon-structed data using the MWNI method (Figure 10c) shows that the signal was reconstructedsuccessfully even though we did not have samples with less than 4 samples apart. This re-flects the fact that it is the main underlying grid of the sampling operator that determinesthe success of the MWNI method. Figures 10b and 10d show the Fourier panels of Figures10a and 10c, respectively.

It is important to mention that for random sampling even relatively high amplitude ar-tifacts can be eliminated using the MWNI method. This is due to the fact that the MWNIalgorithm is a non-linear interpolation method which iteratively updates its weighting func-tion in the wavenumber domain by comparing the latest result of the reconstruction to theavailable samples. Hence, if there were high amplitude artifacts in the spectrum that donot match the available samples, they are eliminated by iterative updating. Unfortunately,artifacts in the spectrum due to regular decimation in the spatial domain will completelymatch the available samples and therefore, non-linear updating can not eliminate them.

Figure 11 shows the relationship between the percentage of available samples and thetotal reconstruction error for the MWNI method when random sampling is adopted to theoriginal data in Figure 9a. For each percentage of missing samples, 20 different randomsampling realizations were performed and the mean of reconstruction error (normalizedRMS difference) and its standard deviation were plotted. As the number of available samplesdecreases, the reconstruction error and its variance increases.

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Figure 11: Reconstruction error vs percentage of available samples for 1D MWNI of randomlysampled data. For each case 20 different random realizations were carried out. The variance isshown by the error bars.

Regular decimation

For the next example (Figure 12a), the signal was decimated by a decimation factor of 2(every other sample). Due to the regularity of the samples, the MWNI method was not ableto reconstruct the missing samples (Figure 12c). While MWNI fails to recover the missingdata, one can utilize band-limitation in the Fourier domain for reconstruction purposes.Figure 12e shows the successful reconstruction of the data using the band-limited MWNImethod. The band-limiting function in the Fourier domain is designed to eliminate anyevent out of the normalized frequencies [−0.15, 0.15]. The interval is chosen based on the apriori information regarding the spectrum of the original data. Figures 12b, 12d, and 12fshow the Fourier panels of Figures 12a, 12c, and 12e, respectively.

While MWNI with the addition of band-limitation can eliminate high amplitude artifactscaused by regular sampling operators, it will fail to do so if the artifacts are mixed with theoriginal spectrum of the data. Figure 13a show a regularly decimated signal with decimationfactor of 6. Figure 13b shows the Fourier panel of Figure 13a. It is evident that parts ofthe artificial events introduced by the regular sampling operator are interfering with theoriginal spectrum of the data. Figure 13c and 13d show the reconstructed data using theband-limited MWNI method and its Fourier panel, respectively. The reconstructed dataare different from the original data due to the presence of high amplitude artifacts in theband selected for reconstruction.

Combination of regular and random sampling operators

A sampling operator can be a combination of regular and random sampling functions.This means that first, a discrete signal is decimated using a regular sampling operator andthen the resultant sampled function is sampled using a random sampling operator. Figure14a shows an example where the available samples have been randomly picked from analready decimated original signal by a factor of 3. Figure 14c shows the reconstructed datausing the MWNI method. Since the random sampling operator is applied on the alreadydecimated signal, the MWNI method was able to reconstruct the decimated signal, notthe original one. Figure 14e shows the reconstructed data using the MWNI method when

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Figure 12: a) Regularly decimated data (decimation factor of 2). c) Reconstructed data via MWNI.e) Reconstructed data using band-limited MWNI in the normalized frequency interval of [-0.15,0.15].b), d), and f) are the wavenumber spectra of (a), (c), and (e), respectively.

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Figure 13: a) Regularly decimated data (decimation factor of 6). c) Reconstructed data using band-limited MWNI in the normalized frequency interval of [-0.15,0.15]. b) and d) are the wavenumberspectra of (a) and (c), respectively.

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Figure 14: a) Random sampling from a regularly decimated data (decimation factor of 3). c)Reconstructed data via MWNI. e) Reconstructed data using band-limited MWNI in the normalizedfrequency interval of [-0.15,0.15]. b), d), and f) are the wavenumber spectra of (a), (c), and (e),respectively.

band-limitation is included. Notice that we have simultaneously utilized band-limitationand the minimum weighted norm constraint to remove the effects of both regular andrandom sampling operators. Figures 14b, 14d, and 14f show the Fourier panels of Figures14a, 14c, and 14e, respectively.

Gap operator

Figure 15a shows an example with a gap inside the data. The reconstructed data usingthe MWNI method (Figure 15c) shows a successful recovery of the samples in the gap.It is clear that the artifacts caused by the gap in the frequency domain (15b) are all inthe vicinity of the main spectrum of the original signal (9b). This was expected since theFourier response of the gap sampling operator (Figure 4) has high value impulses only inthe vicinity of the main impulse of the original spectrum. Therefore, as the Fourier panel ofthe reconstructed data (Figure 15d) shows, the MWNI method was capable of eliminatingthe artifacts caused by a gap inside the regularly sampled grid of data.

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Synthetic 2D examples

Random sampling

In this section we utilize the MWNI method to reconstruct 2D data. In fact, the 2D MWNImethod is carried out in the f-x domain and, therefore, it is implemented as a 1D spatialreconstruction process for each frequency. Figure 16a shows an original synthetic 2D sectioncomposed of three linear events. The original data are severely aliased and the band-limitedMWNI method can not be used for the frequencies above the normalized frequency 0.12.More than 60 percent of traces are randomly eliminated creating a section with missingtraces shown in Figure 16b. Figure 16c shows the reconstructed data using the MWNImethod at each frequency. Figures 16d, 16e and 16f show the f-k spectrum of Figures16a, 16b and 16c, respectively. The random sampling in the spatial domain creates lowamplitude artifacts around the spectrum of the original data. This is the behavior expectedfrom the random sampling operator as shown in Figure 2. In the case of random samplingof traces, the MWNI method is capable of interpolating the missing traces (Figure 16c).However, there are some isolated high amplitude artifacts left in the f-k spectrum of thereconstructed data.

Regular sampling

The same 2D synthetic section was also decimated by a factor of 2 (Figure 17a) in orderto be reconstructed by the MWNI method. Since the sampling operator was regular, theMWNI method was not able to recover the missing traces (Figure 17b). Figures 17c and17d show the f-k spectrum representation of the figures 17a and 17b, respectively. Theeffect of regular sampling function on the spectrum of the data is the superposition of the

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Figure 16: 2D data in the t-x domain. a) Original data. b) Randomly sampled data. c) Recon-structed data via MWNI. d), e), and f) are the f-k spectra of (a), (b), and (c), respectively.

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Figure 17: a) Regular decimation of data in Figure 17a. b) Reconstructed data via MWNI. c) andd) are the f-k spectra of (a) and (b), respectively.

cyclic shifted spectrum with the original one creating the wrapping of the signal in thef-k domain (Gulunay, 2003). Therefore, since the original spectrum of data was spatiallyaliased, the resulted replicated spectrum completely mixes with the original spectrum. Ap-plying the MWNI method separately for each frequency can not discriminate between theoriginal and the replicated spectrum. The replicated spectrum is well-separated in the lowfrequency portion of the data and a band-limiting (low-pass) function can be deployed inthe wavenumber axis to eliminate them. However, high frequency spatial data, the band-limitation is no longer effective. To overcome this problem, Gulunay (2003) introduced anf-k reconstruction method where a mask function is built from low frequencies to eliminatethe aliasing artifacts in the high frequencies.

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Figure 18: a) Data with gap inside the original data in Figure 17a. b) Reconstructed data viaMWNI. c) and d) are the f-k spectra of (a) and (b), respectively.

Gap sampling

Next, we investigate the spatial gap of traces inside a 2D synthetic example (Figure 18a).As it was expected and shown in Figure 18c, the gap causes low amplitude repetitionof the spectrum in the vicinity of the original spectrum. Figures 18b and 18d show thereconstructed data using the MWNI method in the t-x and f-k domains, respectively. Theresults show that the MWNI method is a good candidate for reconstruction of a gap insidethe data.

23

3D examples

Synthetic linear events

In order to examine the performance of the MWNI method for multidimensional data, a 3Dcube of synthetic data was created. The original data are composed of three dipping planeswith slopes (αx, αy) = {(2.5, 3.75), (1.25,−1.25), (−3.5,−3.0)}. The data set contains 2400traces in a spatial grid of 40× 60 and slightly contaminated with random noise. Figure 19ashows a perspective view of the cube containing the synthetic data. The trajectory linesof the three synthetic events can be seen on the boundaries of the cube. In order to getan in-depth view of the data one can pick slices of data in different locations and projectthem to the sides of the cube. Figure 19b shows a cube of the original data where thetop view is the time slice at 0.65 (s), the front view is the 21st slice in the Y direction,and the side view is 17th slice in the X direction. Figure 19c shows the cube with missingtraces on which about 95% of traces have been eliminated randomly. Figure 19d shows thereconstructed data using the MWNI method. Despite the large number of missing traces,the reconstruction was successful. Figures 20a, 20b, and 20c show the f-k panels of thefront views in Figures 19b, 19c, and 19d, respectively.

Figure 21a shows a cube with missing traces which was created from Figure 19b by firstdecimating every other slice in the X direction and then eliminating 50% of the remainingtraces. Overall, in Figure 21a, about 75% of the original traces were eliminated. Figure 21bshows the reconstructed cube using the MWNI method. It is interesting to note that onlyevery other slice in the X direction is successfully reconstructed. This was expected sincethe original decimation of slices in X direction results in a 2D Fourier response similar tothe one shown in Figure 6d. This means that the replicated artifacts in the X direction areas large as the amplitudes in the original spectrum and cannot be removed by the MWNImethod. Since the original data was aliased in both spatial directions we could not use theband-limited MWNI method to eliminate the artifacts. Figures 22a and 22b show the f-kpanels of the front views in Figures 22a and 22b, respectively.

Synthetic curved events

It is interesting to examine the performance of Fourier reconstruction for non-linear (curved)events. Events with curvatures in the t-x domain do not have a simple or sparse represen-tation in the frequency-wavenumber domain. New representation methods like curvelets(Candes et al., 2005) and seislets (Fomel, 2006) were recently introduced to obtain a sparseand local representation of seismic events. This paper, however, is focused on reconstructionmethods based on Fourier transforms. Interested readers are referred to Hennenfent andHerrmann (2008); Naghizadeh and Sacchi (2010) for curvelet-based reconstruction and Liuand Fomel (2010) for seislet interpolation.

Figure 23a shows a synthetic cube of curved events which are aliased in the X directionand alias-free in the Y direction. Figure 23b shows slice views of the original data at time0.35 (s) (top view), X slice 17 (side view), and Y slice 21 (front view). Figure 23c showsdata after eliminating every other slice of data in the X direction. The sampling operatoris equivalent to the one depicted in Figures 6c. Figure 23d shows the reconstructed datausing the band-limited MWNI method by applying band-limitations on the normalized

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Figure 19: a) 3D synthetic cube of data. b) Slice views from inside the original data. c) Data with95% randomly missing traces. d) Reconstructed data using MWNI method.

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Figure 21: a) Cube of missing traces created from the original data shown in Figure 19b by firsteliminating every other X slice and then randomly elimination 50% of the remaining traces. b)Reconstructed data using MWNI.

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Figure 22: a) and b) are the f-k spectra of the front views of the data in Figures 21a and 21b.

wavenumber in the Y direction (ky ∈ [−0.15, 0.15]). The reconstruction was not successfulbecause the sampling artifacts were inside the band-width used by the inversion. Figure23e shows the cube of data with missing traces which has been sampled in a chessboardpattern. The sampling operator is equivalent to the one in Figure 6e. Figure 23f showsthe reconstruction of the data in Figure 23e using the band-limited MWNI method in thebandwidth ky ∈ [−0.15, 0.15]. For the chessboard pattern, the artifacts fall outside thechosen band-width and as a result the band-limited MWNI method can recover the missingtraces successfully. Figures 24a, 24b, and 24c show the f-k panel of front views in Figures23b, 23e, and 23f, respectively.

Interpolation of 3D real data

2D spatial random sampling

A real data set from the Gulf of Mexico was selected to analyze the performance of theMWNI method under different sampling scenarios. A data cube of nineteen consecutiveshots, each with 91 traces, each trace having 900 time samples, is used for this specifictest. Figure 25a shows a perspective view of the cube containing the selected real data withtrajectories of the events on the boundaries of the cube. Figure 25b shows slice views ofFigure 25a at time 3.5 (s) (top view), X slice 46 (side view), and Y slice 10 (front view).Next 50% of the original traces were eliminated randomly to create a cube with missingtraces (Figure 25c). Figure 25d shows the reconstructed data using the MWNI method.Figures 26a, 26b, and 26c show the f-k panels of the front views in Figures 26b, 26c,and 26d, respectively. Because of the presence of curved events there is some differencesbetween the original data and the reconstructed data. Also, notice that the reconstructeddata have been muted above the first arrival. By choosing a proper windowing strategy inthe spatial and time directions, one can further reduce the differences between the original

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Figure 23: a) 3D synthetic cube of data with mild curvature in the Y direction and alias in the Xdirection . b) Slice view from inside the original data. c) Data after elimination of every other slicein the X direction. d) Reconstruction of (c) using band-limited MWNI. e) Data after chessboardelimination of traces. f) Reconstruction of (e) using band-limited MWNI.

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and reconstructed data.

2D spatial regular sampling

Figure 27a shows the cube of data from the Gulf of Mexico on which the original traces aredecimated by a chessboard pattern. The 2D sampling operator for this example is similarto the one in Figure 6e. Figure 28a shows the f-k spectrum of the front view of Figure 27a.It is clear that regular sampling resulted in an f-k spectrum which is the sum of wrapped(shifted) spectrum with the original spectrum. Also there is a prominent overlap betweenthe original and shifted spectrum. Figure 27b shows the reconstructed data using the MWNImethod. Figure 28b shows the f-k spectrum of the front view of Figure 28b. The MWNImethod was not able to eliminate the artifacts. Figure 27c shows the reconstructed datausing the band-limited MWNI method. The band-limitation was applied in the interval ofky = [−0.2, 0.2] normalized wavenumbers and full-band on kx axis. The band-limitation onthe Y direction (common-offset direction) makes sense since one would not expect strongdips as well as sharp changes in this direction. The reconstructed data still contains someleftover artifacts. This could be due to the presence of noise and curvature in the originaldata. Figure 28c shows the f-k panel of the front view of Figure 28c.

CONCLUSIONS

In this paper we examined the effects of random sampling and regular sampling operatorson the performance of the MWNI method. The spectrum of sampling operators and theirinfluence on the original spectrum of the data were mathematically derived. The spectrumof the random sampling operators has only one impulse and the rest of its components are

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small. Conversely, for regularly sampled data, depending on the decimation factor, severalequally sized impulses will be present in the spectral domain. The latter is responsiblefor spectral mixing in the form of alias. The spectrum of the sampling operator is cycliclyconvolved with the spectrum of original data to give the spectrum of sampled data. Providedthat the original data had a simple spectrum, its convolution with the spectrum of therandom sampling operator will still create a simple signal with small amplitude artifacts.Therefore, randomly sampled signals are recoverable by assuming simplicity in the Fourierdomain. The latter can be in the form of sparsity or via the minimum weighted normconstraint studied in this paper.

The spectrum of the regular sampling operator contains replications and its cyclic con-volution with the spectrum of original data will produce unwanted repetitions of the spec-trum of original data. Hence, eliminating the artifacts caused by the regular samplingoperator using the MWNI method will be impossible unless the original data were band-limited. For the multidimensional case (more than one spatial dimensions), the successof the band-limited MWNI method will depend on the type of regular sampling operatorand band-limited properties of the original data. For a chessboard pattern decimation, thesampling artifacts in the Fourier domain will have the optimal possible separation from thespectrum of original data. This property helps one to use the band-limitation informationin one spatial direction to recover the signal in other, aliased, spatial directions.

ACKNOWLEDGMENTS

This research was funded by the generous support of the industrial members of the SignalAnalysis and Imaging Group (SAIG) at the University of Alberta and by a Discovery Grantfrom the Natural Sciences and Engineering Research Council of Canada to M.D. Sacchi. We

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Figure 27: a) Decimation of data in Figure 25b with a chessboard pattern. b) Reconstructeddata using MWNI. c) Reconstructed data using the band-limited MWNI method in the interval ofky = [−0.15, 0.15].

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also thank inspiring interpolation discussions with Ray Abma, Liu Bin, Mike Perz, DanielTrad and Juefu Wang.

APPENDIX

We call Q the wavenumber response of the sampling operator. It can be proven thatXs = X∗Q, where ∗ represents the convolution operator. Writing the convolution operatorexplicitly, we have:

Xs(k) =N−1∑j=0

X(j)Q(k − j)

=N−1∑j=0

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x(n)e−i2πnj

N ][1N

M−1∑m=0

e−i2πh(m)(k−j)

N ]

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[N−1∑n=0

x(n)e−i2πnj

N ][1N

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e−i2πh(m)k

N ei2πh(m)j

N ]

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M−1∑m=0

x(n)e−i2πh(m)k

N [1N

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e−i2πnj

N ei2πh(m)j

N ]. (17)

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Now due to the orthogonality of Fourier series, the expression inside the bracket in the lastline of (17) is nonzero if n = h(m). This means that expression (17) can be reduced to:

Xs(k) =M−1∑m=0

x(h(m))e−i2πh(m)k

N [1N

N ]

=M−1∑m=0

x(h(m))e−i2πh(m)k

N k = 0, 1, 2, . . . , N − 1, (18)

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