A hybrid approach to seismic deblending: when physics meets self-supervision
Abstract
To limit the time, cost, and environmental impact associated with the acquisitionof seismic data, in recent decades considerable effort has been put into so-calledsimultaneous shooting acquisitions, where seismic sources are ﬁred at short timeintervals between each other.
As a consequence, waves originating from consecutive shots are entangled within the seismic recordings, yielding so-called blendeddata.
For processing and imaging purposes, the data generated by each individualshot must be retrieved.
This process, called deblending, is achieved by solvingan inverse problem which is heavily underdetermined.
Conventional approachesrely on transformations that render the blending noise into burst-like noise, whilstpreserving the signal of interest.
Compressed sensing type regularization is thenapplied, where sparsity in some domain is assumed for the signal of interest.
Thedomain of choice depends on the geometry of the acquisition and the propertiesof seismic data within the chosen domain.
In this work, we introduce a newconcept that consists of embedding a self-supervised denoising network into the
Plug-and-Play (PnP) framework.
A novel network is introduced whose designextends the blind-spot network architecture of [27] for partially coherent noise(i.e. , correlated in time).
The network is trained directly on the noisy input data ateach step of the PnP algorithm.
By leveraging both the underlying physics of theblending operator and the great denoising capabilities of our blind-spot network,the proposed algorithm is shown to outperform an industry-standard method whilstbeing comparable in terms of computational cost.
Moreover, being independent onthe acquisition geometry, our method can be easily applied to both marine and landdata without any signiﬁcant modiﬁcation.
Introduction
Reﬂection seismology [41] is a geophysical technique that uses reﬂected seismic waves to characterizethe Earth’s subsurface.
It comprises of a controlled source of seismic energy and an array of receiversthat record the pressure (or displacement) induced by the reﬂected waves.
After the introduction of3D seismic [11], today’s conventional seismic acquisition campaigns may last several weeks up toa few months [7, 25, 10].
In an attempt to improve acquisition efﬁciency, and therefore limit thetime, cost, and associated environmental impact, [6, 9, 3, 36] introduced a new paradigm in seismicacquisition referred to as simultaneous shooting .
Simply put, consecutive sources are ﬁred at shorttime intervals, thereby minimizing the overall acquisition time.
This comes at the cost of recordingentangled seismic data, also called blended data, where the waves originating from one source tend tooverlap with those originating from previous and subsequent sources.
To render such data suitable forsubsequent steps of seismic processing and imaging, the interference between consecutive shots mustbe removed such that the contribution of each individual source (also referred to as a shot gather )is retrieved.
This process is called deblending .
In theory, deblending can be achieved by solving an inverse problem; however, as this problem is heavily underdetermined, choosing an appropriateregularization is fundamental to achieve a successful inversion.
Historically, the design of suitableregularizers is motivated by the effect of the adjoint of the blending operator on the blended data.
Infact, the resulting data can be seen as a superposition of coherent signal (i.e, reﬂections from the shotwhose ﬁring time has been properly accounted for) and trace-wise, burst-like noise (i.e, reﬂectionsfrom all other interfering shots whose ﬁring times have not been properly accounted for).
The recent success of deep learning in various scientiﬁc disciplines has attracted the interest of thegeophysical community, resulting in many opportunities and new challenges [51].
For example,whilst training data should consist of clean, representative ground truth examples that resemblethe solution to the inverse problem at hand, such data is generally not available.
Two approachescommonly adopted to circumvent this problem are to either generate synthetic data or to use stateof-the-art algorithms to produce input-output pairs to train a network on; in both cases, transferlearning [42, 34] or domain adaptation [2, 12] techniques are then required to generalize the networkcapabilities to unseen ﬁeld data.
A major drawback of the ﬁrst approach is that synthetic data may notresemble ﬁeld data accurately enough to be considered a representative dataset: this is well-knownin the geophysical community and has been a major criticism for decades when new methods aretested only on synthetic data.
It also represents a serious roadblock to the application of deep learningmethods in geophysics.
Additionally, in most geophysical applications the underlying physics is(at least partly) well understood.
Pure, end-to-end machine learning methods tend to ignore thesewell-studied physical principles, thereby discarding important a priori knowledge of the problem theyare tasked to solve.
Our contribution We introduce a novel algorithm for seismic deblending, which combines thephysics of the underlying physical process with a state-of-the-art self-supervised denoiser into asingle, well-crafted inverse process.
This is speciﬁcally achieved within the framework of Plugand-Play (PnP) priors.
Our network architecture is inspired by the blind-spot network of [27] andmodiﬁed to handle trace-wise coherent noise.
The network is trained on-the-ﬂy at each PnP iterationin a self-supervised manner, completely bypassing the need for ground truth data.
Our numericalexperiments illustrate that the proposed algorithm can outperform a state-of-the-art conventionalmethod.
Finally, we show that our algorithm is independent on the underlying structure of the seismicdata and can be used easily for different acquisition set-ups - a clear advantage over conventionalmethods.
Background
The seismic data layout Seismic data are commonly acquired by ﬁring a source at a given time andrecording the reﬂections arising from the interaction between the emitted seismic wave and changesin subsurface properties.
Conceptually, seismic data can be arranged as a three dimensional tensor (ora cube), having the dimensions of the number of sources n s , number of receivers n r , and number oftime samples n t : d c ( x s , x r , t ) .
Slicing this cube in different directions gives raise to so-called seismicgathers: more speciﬁcally, when slicing across the source axis, we obtain the data recorded by allreceivers for a single shot, usually called Common Shot Gather (CSG); conversely, by slicing acrossthe receiver axis we obtain the data generated by all shots for a single receiver.
When the receiversmove alongside the source (i.e., marine case) the resulting gather is called Common Channel Gather(CCG).
For static receivers (i.e., ocean-bottom or land acquisition), the seismic gather is know as the Common Receiver Gather (CRG).
Both scenarios will later be considered.
Blended acquisition In practice, to be able to collect data where no overlap exists betweenconsecutive shots, each shot has to be ﬁred with an appropriate time delay, such that all reﬂectionsfrom one shot have been recorded by the receivers before the next shot is ﬁred.
This dictatesthe overall acquisition time and greatly limits any possible acquisition speed-up.
Alternatively,in blended acquisition, shots are ﬁred at shorter intervals.
This means that each individual CSGcontains recordings from both the nominal as well as the previous and subsequent shots.
In this work,we consider the so-called continuous blending setting.
This approach is state-of-the-art in marineseismic acquisition due to the fact it is easy to implement in the ﬁeld.
It is achieved by ﬁring theairgun towed by the acquisition vessel at short time intervals, and continuously recording the wavesreturning to the receiver array as depicted in ﬁgure 1.
The recorded data d b can be simply describedas the superposition of all of the unblended, or clean, data shifted in time by the given time delay t i = i · T +  t i .
Here, T is the nominal ﬁring interval and  t i is a random dither applied to the 92 nominal ﬁring time of shot i .
The blended data can thus be described as a function of the clean data
[Equation]
where the blending operator is a horizontal stack of time-shift operators B i , and the clean data is avector where all vectorized shot gathers, d c,i = vec ( d c ( x s,i , x r , t )) , are stacked together.
Moreover,each B i time-shift operator has the property that B Ti B
i
=
I , and a composition of time-shift operatorsis again a time-shift operator [33].
Pseudo-deblending To better understand how to design effective regularization strategies for thedeblending problem, we ﬁrst have to consider the action of B H on the blended data.
For the i th shotgather, the result of B Hi Bd c can be written as
[Equation]
Therefore the action of B Hi on the blended data produces the original i th shot gather alongsiderandomly shifted versions of all the other shot gathers.
Whilst these randomly shifted shot gathersare coherent and look like standard seismic signal in the CSG domain, they appear as trace-wisecoherent noise in the CRG (or CCG) domain as shown in ﬁgure 1(c).
Because the application of theadjoint of the blending operator retrieves the true signal, albeit with some additional noise, its actionis usually called pseudo-deblending .
As a consequence of this, it is now clear that to retrieve thevarious d c,i , an effective regularization must ﬁlter the trace-wise noise in CRGs (or CCGs) whilstpreserving the coherent signal.
Conventional approaches identify a domain in which the signal can beeasily discriminated from the noise, and more speciﬁcally the signal in such domain is sparse whilstthe noise is not.
Examples of such a kind include the hyperbolic Radon transform for CRGs [24], thepatched Fourier transform for CCGs [1], or the Curvelet transform [30].
Deblending by inversion Deblending by denoising is achieved by minimizing
[Equation]
whereas deblending by inversion amounts to retrieving the clean data by solving the (heavily)underdetermined inverse problem,
[Equation]
where R ( · ) is any chosen regularization.
The literature has shown that deblending by inversion is 115 superior to deblending by denoising in terms of the overall quality of reconstruction and will be thefocus of this work.
More details on both approaches are provided in the supplementary material. Self-supervised denoising: Incoherent noise Self-supervised denoisers are designed in such a waythat noisy images can be used as both the input and label to train a neural network to act as a denoiser,thereby bypassing the need for clean data as labels.
Noise2Noise represents the ﬁrst such methodnot relying on ground truth labels [28].
The network is forced to infer the signal from pairs of noisydata.
For applications where such pairs are unavailable, an alternative was proposed in the concurrentworks of [26] and [5], who introduced Noise2Void and Noise2Self, respectively.
In both cases, thesame image is used as input and label: under the assumption that the noise is incoherent whilst thesignal is coherent, the network can naturally learn to infer only the signal from its neighbouringpixels.
More speciﬁcally, to denoise a particular pixel, [26] replace the pixel of the input image witha randomly selected neighbouring pixel.
As this pre-processing step introduces randomness in thecentral pixel of the receptive ﬁeld of the network, the network should not learn anything from it andnaturally learns to infer the signal from its neighbours (since the noise is assumed to be incoherent).
Rather than directly replacing the pixel of interest, [5] pre-process the input image with a blind-spotconvolutional ﬁlter, so that the network cannot rely on the central pixel to predict itself.
A keylimitation of both approaches lies in the fact that the self-supervised loss can be evaluated only at thepixels that have been corrupted, making the training of these denoisers relatively slow.
An alternativeapproach to blind-spot networks was introduced by [27].
Instead of corrupting the middle pixel, theirnetwork is explicitly designed to have a receptive ﬁeld with a hole in the middle.
This is achieved bycombining padding and cropping with a standard convolution layer (i.e., to create a causal ﬁlter) andby rotating the input image four times prior to feeding it through the network.
After the rotated inputshave been fed through the network, they are rotated back, concatenated, and combined by a seriesof 1 ⇥ 1 convolutions prior to evaluating the loss at every pixel of the output image.
A schematic 139 description of this network is depicted in ﬁgure 2a.
Self-supervised denoising: Coherent noise Both Noise2Void and Noise2Self operate under the 141 assumption that the noise is independent and identically distributed.
[15] shows that the denoisingquality of Noise2Void is degraded when the noise is structured.
This shortcoming of Noise2Void issolved by masking pixels along the direction of the noise: the authors dub their method Structured
Noise2Void.
For the seismic deblending problem, the noise that we are interested to suppress is alsostructured: more speciﬁcally, the blending noise shows correlation along the time axis.
We, therefore,extend here the efﬁcient implementation of [27] to suppress structured noise in seismic data, by usingthe original and ﬂipped (over the source axis) version of the image as input.
This produces a networkwhose receptive ﬁeld is masked over an entire time trace, see ﬁgure 2b.
In the following, we will callthis network Structured Blind Spot, or StructBS for short.
Related work
Simultaneous shooting Simultaneous shooting was ﬁrst pioneered by [6] and has gained popularityin recent years [9].
Although the ﬁrst attempts at deblending were mostly by means of denoising [33],recent research has reveled the superiority of deblending by inversion [1].
Since then, research hasbeen devoted to ﬁnding appropriate regularization terms.
Some approaches involve median-ﬁltering[22, 21, 23], rank-reduction methods [17, 55], sparse regularization [29, 30, 56, 57, 59], and deeplearning [43, 58, 49].
All the deep learning approaches to date use CNNs and require pre-training.
[4]uses the RED framework introduced in [38], which is similar to the PnP framework.
The differenceis that RED explicitly incorporates the denoiser into the objective function.
The authors propose theuse of two conventional regularization techniques as a denoiser, the patched Fourier transform [1]and the singular-spectral-analysis ﬁlter [17], instead of applying them as a sparse penalty.
Self-supervised seismic denoising Seismic data are a prime example of a noisy data type whereno clean, ground truth labels are available.
As such, the application of self-supervised denoisers hasrecently been proposed for the suppression of different types of noise present in seismic data.
Following the Noise2Void methodology, [13] use blind-spot networks for the suppression of random noisein post-stack seismic data.
Expanding on this, [32] adapted the methodology of StucturedNoise2Void[15] for the suppression of trace-wise noise in seismic shot gathers, originating from poorly coupledreceivers and/or dead sensors.
The method that is most closely related to ours is the one presented in[49] - both with respect to application and methodology.
The authors propose to use a self-superviseddenoising network to deblend the data by denoising.
To produce satisfactory results they require anumber of additional pre- and post-processing steps.
In our work, we incorporate a deep learningbased denoiser in deblending by inversion, thereby leveraging both the underlying physics and thepower of neural networks.
Moreover, no pre- and post-processing is required.
The Plug-and-Play framework The Plug-and-Play framework was pioneered by [48].
The authorsconsidered a number of popular denoisers, including BM3D [18], K-SVD [19], PLOW [16] and qGGMRF [45].
In subsequent works, the denoisers have been replaced by pre-trained neural networks,most notably CNN and DnCNN.
Lately, [31] proposed regularization by artifact-removal (RARE),a method leveraging a Noise2Noise type approach that requires pre-training.
An extensive list ofreferences is provided in [52], and include [38, 54, 35, 47, 20, 46, 29, 44, 53].
This research focuseson progressively training a neural network such that it can adapt to changing noise levels.
The noveltyof our method is that pre-training is not required.
Method
Equipped with a self-supervised denoiser, a straightforward approach to deblending is to directlydenoise the pseudo-deblended data.
However, deblending by denoising is known to be sub-optimal incomparison to deblending by inversion.
On the other hand, because a denoiser cannot be naturallyadded as a constraint to the objective function in equation 4, it is not immediately clear how toincorporate the denoiser into the inversion process.
[48] proposed the PnP framework, which isdirectly derived from the Alternating Direction Method of Multipliers (ADMM).
Whilst resemblingthe alternating minimization process of the classical ADMM algorithm, PnP is more ﬂexible inthe sense that it can use any denoiser of choice, without the need for it to be linked to an explicitregularization term for the so-called y -update. To understand our method clearly, we give a shortderivation of the ADMM following [14], we then link it to the PnP algorithm and ﬁnally to ourproposed algorithm. The ADMM algorithm is generally used to solve inverse problems of the form [Equation]
where M is the forward model, d is the measured data, D is a data ﬁdelity term that is generally 194 smooth, and R is a convex, possibly non-smooth regularization term. Due to the non-smoothness 195 of the objective, this problem cannot be solved with standard gradient-based methods. To account for the non-smoothness of R , an auxiliary variable y = x is introduced, yielding the equivalent 197 optimization problem
[Equation]
to
[Equation]
y
[Equation]
ADMM solves this problem by forming the so-called augmented Lagrangian ,
[Equation]
where u is the Lagrange multiplier and ⇢ is a scalar. This problem is solved by alternativelyminimizing over x and y , and maximizing over u . This yields the following scheme: [Equation]
The introduction of y = x and the addition of the quadratic penalty ⇢ 2 k x  y k yields the y -update, 202 which for most popular regularization terms has a simple closed-form solution that can be cheaplyevaluated [37]. The key observation of [48] is that the y -update can be interpreted as a denoisinginverse problem. As such, the authors propose to drop the user-deﬁned regularization R ( · ) and 205 instead plug in a denoiser of choice in the y -update of the ADMM iterations.
Although this may notseem a straightforward choice, PnP has been shown to be competitive (or sometimes even better) thanstandard regularization methods in a variety of settings. Given the trace-wise structure of the noiseand equipped with the self-supervised denoiser, the PnP framework becomes a natural and attractivechoice for the deblending task at hand. Our proposed algorithm reads as follows: [Equation]
where x is used here for simplicity in place of d c , and the x -update is performed using an iterativesolver of choice, e.g. LSQR.
The y -update is now the denoiser StructBS ✓ , where ✓ denote the networkparameters.
The variable u couples both x
and y and forces them to be close together.
The x -updaterequires the solution to satisfy the physics dictated by the equation Bx = d b , and the y -updatedenoises the noisy receiver gathers.
Experiments
In the following, our algorithm is tested on the openly available Mobil AVO viking graben line12 marine dataset 1 .
As the data has been originally acquired in a conventional fashion, we createthe blending operator and blend the data ourselves.
In addition to containing all the challengingfeatures of a ﬁeld dataset, this also provides us with a ground truth, d c , onto which to assess thequality of our reconstruction.
In this example, the original dataset is composed of n s = 64 sources, 221 n r = 120 receivers, and n t = 1024 samples (i.e., the total recording time per shot equals 4 seconds).
For the continuous blending operator, we choose a ﬁxed ﬁring interval of T = 2 seconds, withadded random delays selected uniformly in the interval  t i ⇠ [  1 , 1] seconds.
This overlap is quite 224 challenging as generally half of the signal overlaps with either that of the previous or that of the nextshot.
Moreover, some pseudo-deblended shot gathers exhibit contributions from three consecutiveshots.
Finally, the relative mean-square error, RMSE =
k d c  d c,true k 2 / k d c,true k 2 , is chosen 227 as a metric of comparison in all of our numerical examples.
All experiments are performed on a
Intel(R)
Xeon(R) CPU @
2.10GHz equipped with a single NVIDIA GEForce RTX 3090 GPU.
5.1 Comparison with state-of-the-art deblending
To begin with, our newly proposed methodology is compared with the state-of-the-art deblendingalgorithm of [1] that solves the deblending problem as a sparsity promoting inversion
[Equation]
where F is a linear operator that performs a patched two-dimensional Fourier transform, and z ? is thesolution in the Fourier domain that is ultimately transformed back to the original time-space domainof the seismic data.
In our experiment, the size and number of patches as well as the regularizationparameter  are selected by hand to yield optimal results.
Moreover, the FISTA [8] solver is used withan adaptive decreasing sequence,  k (as this has been shown in the literature to outperform a ﬁxed for this speciﬁc problem).
We choose the sequence  k =  65 e  0 .
k +  0 , which was once againﬁne-tuned to give the best performance.
The ﬁnal error is roughly 9 .
8% (see supplementary materialfor details); hereon in this represents the benchmark against which we will assess the effectivenessof our self-supervised PnP algorithm.
Next, our PnP algorithm is applied to the same dataset.
Wechoose 30 outer iterations, 3 inner iterations, ⇢ = 1 and 30 denoiser epochs.
The choice of thesehyperparemeters will be justiﬁed in the ablation study.
We also use the U-Net architecture in [27],the L 1 norm for the self-supervised training loss because it is more appropriate for burst-like noise,and the Adam optimizer with default parameters.
Since the denoiser is trained on all the CCGs,the size of our training data is 120 and we use a batch size of 8.
This leads to a solution that hasan overall error of roughly 6 .
7% , which is approximately 3% lower than the conventional method.
As a visual comparison, ﬁgure 3 displays the results for a given CCG (top) and CSG (bottom) forboth the conventional and proposed approaches.
It is noteworthy that our algorithm shows a clearimprovement in terms of denoising capabilities, as visible in the displayed CCG.
Especially after 250 t = 2 s , where the signal is weak and blending noise dominates, the conventional approach tends to bemore prone to signal leakage compared to our PnP algorithm.
Finally, the computational cost of theconventional algorithm can be quantiﬁed in terms of the number of forward and adjoint operationsfor both the blending ( B ) and patched Fourier ( F ) operators: in our example, this amounts to 200forward and adjoint passes.
On the other hand, our method requires 90 forward and adjoint passesfor the blending operator and a total of 900 training epochs for the network.
Considering that allcomputations (apart from the network related ones) are performed on the CPU, the two algorithmsare comparable in terms of overall computational time (2h and 34mins for the conventional algorithmand 1h and 51mins for the PnP algorithm).
5.2 Ablation study
This section provides an extensive analysis of some of the key components of the proposed PnPmethodology and their impact on the overall solution of the deblending inverse problem.
PnP iterations To begin with, we assess the importance of the PnP iterations compared to simplytraining the self-supervised denoiser on pseudo-deblended data and applying it directly to the entiredataset.
Although not shown here, the result of this one-shot denoising produces a solution with anoverall error of roughly 19% .
This is much worse than both the conventional and PnP method andtherefore considered not suitable.
The x -update The ablation study with regard to the x -update is provided in the supplementarymaterial.
This includes a study on the effect of the number of inner iterations and the ⇢ parameter.
Forour continuous deblending problem, we have shown that they could be safely ﬁxed to 3 and 1.
Futureexperiments with different datasets and blending strategies are required to verify this assumption.
The y -update
In our implementation we propose to start with a randomly initialized network andtrain it for a ﬁxed number of epochs at every outer iteration.
A warm start strategy is employed suchthat the weights of the network at a given outer iteration are initialized to those obtained at the endof the training of the previous outer iteration.
The efﬁcacy of this approach is shown in ﬁgure 4a,where we compare on-the-ﬂy training with and without warm starts, where the latter re-initializesthe network at every y -update.
From the error curves, we can safely conclude that warm starting thenetwork is clearly beneﬁcial.
Since there is no theoretical justiﬁcation for this particular strategy, weconsider a few other alternative strategies.
The ﬁrst strategy is to use a pre-trained network.
Herepre-training is achieved by denoising the pseudo-deblended data in a self-supervised manner; thisapproach could greatly reduce the computational cost of the overall algorithm since we do not needto train the network at every iteration.
A comparison of the relative error with that of the proposed,on-the-ﬂy training shown in ﬁgure 4b reveals that after a few outer iterations, the network is unableto further remove the remaining noise in the data.
algorithm with a network pre-trained on the pseudo-deblended data.
Another option is to stop training the network after a few outer iterations.
Ideally, the network willhave learnt how to remove the noise encountered during the ﬁrst iterations, and extra training willnot improve the denoiser capabilities.
We run experiments where we stop the training after a ﬁxednumber of outer iterations to see whether there is an added beneﬁt to continuing training the network.
Results are shown in ﬁgure 5a.
In all of the scenarios we clearly see that stopping the training after
289 a certain number of outer iterations leads to a stagnation in the error, or even worse to an increase in the error at later iterations.
This behaviour is known as semiconvergence in the inverse problemscommunity.
Both results are perhaps not surprising, as the input to the network at every iterationcontains a different noise level compared to that of earlier iterations: the noise in x k constantlyreduces during the overall inversion.
Therefore, the network is required to learn a slightly differenttask at each time.
In ﬁgure 5b, we assess the impact of the number training epochs for the denoiser.
We clearly observe that at some point the curves for 30 and 20 epochs start to coincide, meaningthat there is no additional gain in the extra 10 epochs of training.
The curve for training with 10epochs seems stagnant after the ﬁrst three iterations, but eventually it picks up momentum and goesdown again.
Note that, in terms of overall epochs, the cost of performing 30 outer iterations with10 epochs each is the same as using 10 outer iterations with 30 epochs each.
However, every outeriteration carries an additional cost of three inner iterations for the x -update, which is not negligible asit requires evaluating the forward and adjoint of the blending operator.
In general, it seems beneﬁcialto perform more epochs in the early outer iterations, although there is a limit after which the errorstarts to stagnate.
Moreover, training with 60 epochs leads to overﬁtting.
Finally, to further investigate the generalization capabilities of the network for blending problems,the network weights are saved after the last outer iteration of the PnP algorithm.
The PnP algorithmis then re-run using the saved network without performing any on-the-ﬂy training.
Figure 5c showsthat this strategy fails, illustrating that the network may have forgotten how to deal with the highernoise levels encountered in the early iterations.
This results highlights the importance of using aself-supervised denoiser that can be easily and cheaply trained on-the-ﬂy.
The use of pre-traineddenoising networks such as DnCNN may instead require training multiple networks with differentnoise levels, unless a bias-free, non-blind network is used [52].
Limitations and Conclusions
Limitations Our algorithm requires the setting of a number of hyperparameters, namely the numberof inner and outer iterations, the parameter ⇢ , and the number of epochs for the self-superviseddenoiser.
The number of epochs seems to have a major impact on the quality of the deblendingprocess and the overall convergence properties of our algorithm.
Additional hyperparameters thathave not been explored in this work are associated with the network itself, e.g. the number of layers,the activation function, batch size, etc.
This is also a direction for further research.
Another drawbackis that there is no convergence guarantee, since our operator B is underdetermined and therefore notstrongly convex [39].
Empirically, we observe that x k and y k tend to converge to similar values forsome carefully selected hyperparameters, indicating that at least in our experiments the algorithmconverges successfully.
Similarly, to obtain convergence guarantees for the PnP method, the denoiserhas to be Lipschitz continuous; when a neural network is used, this means that spectral normalizationis required during training.
In [50], it was shown that PnP algorithms can be convergent whencombined with carefully pre-trained denoisers that satisfy such condition.
Societal impact Blended acquisition greatly reduces the time required to acquire seismic data,thereby limiting the impact of seismic acquisitions on the environment.
Apart from shooting at shorterintervals, there is no difference compared to conventional acquisition.
Moreover, recent research hassuggested that the energy emitted by each source could be lowered.
This may provide acquisitionsolutions that are more environmentally friendly for marine life.
Conclusions We have introduced a novel hybrid algorithm for seismic deblending, combining thephysics of the blending operator with a self-supervised denoiser that is naturally embedded into the
Plug-and-Play framework.
We have adapted the network architecture in [27] to enforce an extendedblind spot along an entire axis (time, in our case) instead of single pixels.
Because the denoiser isself-supervised, our approaches bypasses the need for ground truth labels that are usually unavailablefor seismic applications.
Experiments on a ﬁeld dataset have shown that the proposed method canoutperform a state-of-the-art, sparsity-based algorithm.
Moreover, as show in the supplementarymaterial, our algorithm is independent on the type of acquisition, which is usually an issue forconventional algorithms.
Although our algorithm requires the setting of a number of hyperparameters,we have argued that the number of inner iterations and ⇢ can most likely be set to a ﬁxed numberand this easily generalizes to different seismic acquisitions.
However, the network architecture andthe number of epochs may require tuning for different acquisition setups.
We hope to address theseissues by having an adaptive strategy for setting the number epochs in future work.