A hybrid approach to seismic deblending: when physics meets self-supervision
Anonymous Author(s)
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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 individual shot must be retrieved.
This process, called deblending, is achieved by solving an inverse problem which is heavily underdetermined.
Conventional approaches rely on transformations that render the blending noise into burst-like noise, whilst preserving the signal of interest.
Compressed sensing type regularization is then applied, where sparsity in some domain is assumed for the signal of interest.
The domain of choice depends on the geometry of the acquisition and the propertiesof seismic data within the chosen domain.
In this work, we introduce a new concept 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 [28] for partially coherent noise (i.e.,correlated in time).
The network is then trained directly on the noisy input dataat each step of the PnP algorithm.
By leveraging both the underlying physics ofthe problem 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 [39] is a geophysical technique that uses reflected seismic waves to characterizethe Earth’s subsurface.
This method comprises of a controlled source of seismic energy and an arrayof receivers that record the pressure (or displacement) induced by the reflected waves.
The physicalinstrumentation varies depending on the acquisition environment: for example, an airgun is used asseismic source in marine settings whilst a vibroseis is the preferred choice in land settings.
Similarly,hydrophones (and possibly accelerometers) are towed behind a boat inside a so-called streamercable or geophones (combined with hydrophones) can be laid directly on the seafloor for marineacquisition; whilst geophones are used for land acquisition.
After the introduction of 3D seismic[11], today’s conventional seismic acquisition campaigns may last several weeks up to a few months[7, 26, 10].
In an attempt to improve acquisition efficiency, and therefore limit the time, cost, andassociated environmental impact, [6, 9, 3, 36] introduced a new paradigm in seismic acquisitionusually 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 recording entangled 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 ) isretrieved.
This process is called deblending .
Mathematically speaking, blending is an operator thattakes the otherwise conventionally acquired data and combines them together using a pre-definedsequence of time delays.
In theory, deblending can be achieved by solving the associated inverseproblem; however, as this problem is heavily underdetermined, choosing an appropriate regularizationis fundamental to achieve a successful inversion.
Historically, the design of suitable regularizersis motivated by the effect of the adjoint of the blending operator on the blended data.
In fact, theresulting data can be seen as a superposition of coherent signal (i.e, reflections from the shot whosefiring time has been properly accounted for) and trace-wise, burst-like noise (i.e, reflections from allother 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 [49].
A common hurdlewhen applying deep learning methods to geophysical inverse problems is the lack of trustworthytraining data.
More specifically, whilst training data should consist of clean, representative groundtruth examples that resemble the solution to the inverse problem at hand, such a solution generallydoes not exist a priori.
Two approaches commonly adopted to circumvent this problem are to eithergenerate synthetic data or to use state-of-the-art algorithms to produce input-output pairs to traina network on; in both cases, transfer learning [40, 34] or domain adaptation [2, 12] techniques arethen required to generalize the network capabilities to unseen field data.
A major drawback of thisapproach is that synthetic data may not resemble field data accurately enough to be considereda representative dataset: this is well-known in the geophysical community and has been a majorcriticism for decades when new methods are tested only on synthetic data.
This also represents aserious roadblock to the application of deep learning methods in geophysics.
Additionally, in mostgeophysical applications the underlying physics , or at least part of the physics, is well understood.
Pure, end-to-end machine learning methods tend to ignore these well-studied physical principles,thereby discarding important a priori knowledge of the problem they are tasked to solve.
Our contribution We introduce a novel algorithm for seismic deblending, which blends the physicsof the underlying physical process with a state-of-the-art self-supervised denoiser into a single, wellcrafted inverse process.
This is specifically achieved within the framework of Plug-and-Play (PnP)priors.
Our network architecture is inspired by the blind-spot network of [28] and modified to handletrace-wise coherent noise.
The network is trained on-the-fly at each PnP iteration in a self-supervisedmanner, completely bypassing the need for ground truth data.
Our numerical experiments illustratethat the proposed algorithm can outperform a state-of-the-art conventional method.
Finally, we showthat our algorithm is independent on the underlying structure of the seismic data and can be usedeasily for different acquisition set-ups - a clear advantage over conventional methods.
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 give 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 a 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 the Common Channel Gather (CCG).
For static receivers (i.e., ocean-bottom or land acquisition), the seismic gather is knowas 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 CSG containsrecordings from both the nominal as well as the previous and subsequent shots.
In this work, weconsider the so-called continuous blending setting.
This approach is state-of-the-art in marine seismicacquisition due to the fact it is easy to implement in the field.
It is in fact simply achieved by firingthe airgun towed by the acquisition vessel at short time intervals, and continuously recording thewaves returning to the receiver array as depicted in figure 1.
The recorded data d b can be simplydescribed as the superposition of all of the unblended, or clean, data shifted in time by the given timedelay t i = i · T + ∆ t i .
Here, T is the nominal firing interval and ∆ t i is a random dither applied tothe nominal firing time of each shot.
The blended data can thus be described as a function of theclean data
[Equation]
where the blending operator is a horizontal stack of time-shift operators B i , and the clean data isa vector where all vectorized shot gathers, d c,i = vec ( d c ( x s,i , x r , t )) , are stacked together.
Notethat each B i time-shift operator has the property that B Ti B
i
=
I , and a composition of time-shiftoperators is again a time-shift operator. .e.
, i · T ), whilst the solid color lines represent the actual firing times with dithering. Color rectangles refer to every individual shot gather that we wish to separate
from the
other overlapping gathers.
c)
Pseudo-deblended data for a single
receiver
(white dashed line in panel b
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 gathers arecoherent and look like standard seismic signal in the CSG domain, they appear as trace-wise coherentnoise in the CRG (or CCG) domain as shown in the Experiments section.
Because the application of the adjoint of the blending operator retrieves the true signal, albeit with some additional noise, itsaction is usually called pseudo-deblending .
As a consequence of this, it is now clear that to retrievethe various 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 are the hyperbolic Radon transform for CRGs [25], thepatched Fourier transform for CCGs [1], or the Curvelet transform [31].
It should be noted here thatthe performance and computational cost of such conventional approaches varies greatly. As explainedlater, a self-supervised denoiser can also be empowered to filter this noise, and, in fact, its denoisingcapabilities will be shown to be superior to those of commonly used, hand-crafted priors.
Deblending by denoising or by inversion In deblending by denoising the clean data is retrievedby directly denoising the pseudo-deblended receiver (or channel) gathers:
[Equation]
where R ( · ) is any chosen regularization.
Because the distribution of the blending noise is far from
Gaussian, a typical choice for the data fidelity norm is ∥ · ∥ 1 . However, the ℓ 1 -norm in the datafidelity makes this problem not trivial to solve. We justify the validity of deblending by denoising inthe supplementary material. Alternatively, one could retrieve the clean data by solving the (heavily)underdetermined inverse problem,
[Equation]
As the incoherent noise is fully explained by the blending operator, the ℓ 2 -norm for the data fidelityis an appropriate choice. From the above equation, it is not directly clear how the noise appears inthe data since we are not dealing directly with the pseudo-deblended data. The noise is howeverintroduced into the solution through the gradient of the data term, B H ( Bd c − d b ) . In sparse inversion,the objective is minimized iteratively by thresholding the gradient at every iteration, which can beinterpreted as a noise removal step.
The literature has shown that deblending by inversion is superiorto to deblending by denoising in terms of the overall quality of reconstruction.
Self-supervised denoising: Incoherent noise Self-supervised denoising is a recently introducedconcept in the deep learning literature. This family of methods are designed in such a way that noisyimages can be used as both the input and label to train a neural network to act as a denoiser, therebybypassing the need for clean data as labels.
Noise2Noise represents the first such method not relyingon ground truth labels [29].
The network is forced to infer the signal from pairs of noisy data.
Forapplications where such pairs are unavailable, an alternative was proposed in the concurrent worksof [27] and [5], who introduced Noise2Void and Noise2Self, respectively.
In both cases, the sameimage is used as input and label: under the assumption that the noise is incoherent whilst the signal iscoherent, the network can naturally learn to infer only the signal from its neighbouring pixels.
Thismechanism bears similarity to classical statistical denoisers such as non-local means [15] or BM3D[18]
, however it relies on a more flexible nonlinear mapping using learned convolutional filters.
Morespecifically, to denoise a particular pixel, [27] replace the pixel of the input image with a randomlyselected neighbouring pixel.
As this pre-processing step introduces randomicity in the central pixel ofthe receptive field of the network, the network should not learn anything from it and naturally learnsto infer the signal from its neighbours (since the noise is assumed to be incoherent).
Rather thandirectly replacing the pixel of interest, [5] pre-process the input image with a blind-spot (or donut)convolutional filter, so that the network once again cannot rely on the central pixel to predict itself.
A key limitation of both approaches lies in the fact that the self-supervised loss can be evaluatedonly at the pixels that have been corrupted, making the training of these denoisers relatively slow.
An alternative approach to blind-spot networks was introduced by [28].
Instead of corrupting themiddle pixel, their network is explicitly designed to have a receptive field with a hole in the middle.
This is achieved by combining padding and cropping with a standard convolution layer (i.e., to createa causal filter) and by rotating the input image four times prior to feeding it through the network.
After the rotated inputs have been fed through the network, they are rotated back, concatenated, andcombined by a series of 1 × 1 convolutions prior to evaluating the loss at every pixel of the outputimage.
A schematic description of this network is depicted in figure 2a.
Self-supervised denoising: Coherent noise Both Noise2Void and Noise2Self operate under theassumption that the noise is independent and identically distributed.
[14] 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 blending 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 [28] to suppress structured noise in seismic data, bysimply using the original and flipped (over the time axis) version of the image as input.
This producesa network whose receptive field is masked over an entire time trace, see figure 2b.
In the following,we will call this network StructBS.
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[23, 22, 24], rank-reduction methods [17, 53], sparse regularization [30, 31, 54, 55, 57], and deeplearning [41, 56, 47].
All the deep learning approaches to date use CNNs and require pre-training.
[4]uses the RED framework introduced in [37], 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[14] 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[47] - 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 [46].
The authorsconsidered a number of popular denoisers, including BM3D [19], K-SVD [20], PLOW [16] and qGGMRF [43].
In subsequent works, the denoisers have been replaced by pre-trained neural networks, most notably CNN and DnCNN.
An extensive list of references is provided in [50], and include[37, 52, 35, 44, 21, 45, 30, 42, 51].
This research focuses mostly on properly training the neuralnetworks such that they generalize well to different noise levels.
The novelty of our method is thatthere is no pre-training with clean data 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.
[46] 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.
Our proposed algorithm reads as follows:
[Equation]
where x is used here for simplicity in place of d c , and the x -update is solved by means of LSQR.
Forthe y -update, we suggest to use the previously introduced structured self-supervised denoiser.
Weprovide a detailed derivation of the PnP method in the supplementary material.
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,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 quitechallenging 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 =
∥ d c − d c,true ∥ 2 / ∥ d c,true ∥ 2 , is chosenas 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 λ are selected by hand to yield optimal results.
Moreover, the FISTA [8] solver is used withan adaptive decreasing sequence, λ k (as this has been shown in the literature to outperform a fixed λfor this specific problem).
We choose the sequence λ k = (cid:0) 65 e − 0 .
05 k + 6 (cid:1) λ 0 , which was once againfine-tuned to give the best performance.
The final error is roughly 9 .
8% (see supplementary material for 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, ρ = 1 and 30 denoiser epochs.
The choice of thesehyperparemeters will be justified in the ablation study.
We also use the U-Net architecture in [28]and the L 1 norm for the self-supervised training loss because it is more appropriate for burst-likenoise.
Since the denoiser is trained on all the CCGs, the size of our training data is 120 and weuse a batch size of 8.
This leads to a solution that has an overall error of roughly 6 .
7% , which isapproximately 3% lower than the conventional method.
As a visual comparison, figure 3 displays theresults for a given CCG (top) and CSG (bottom) for both the conventional and proposed approaches.
It is noteworthy that our algorithm shows a clear improvement in terms of denoising capabilities, asvisible in the displayed CCG.
Especially after t = 2 s , where the signal is weak and blending noisedominates, the conventional approach tends to be more prone to signal leakage compared to our PnPalgorithm.
Finally, the computational cost of the conventional algorithm can be quantified in termsof the number of forward and adjoint operations for both the blending ( B ) and patched Fourier ( F )operators: in our example, this amounts to 200 forward and adjoint passes.
On the other hand, ourmethod requires 90 forward and adjoint passes for the blending operator and a total of 900 trainingepochs for the network.
Considering that all computations (apart from the network related ones) areperformed on the CPU, the two algorithms are comparable in terms of overall computational time (2hand 34mins for the conventional algorithm and 1h and 51mins for the PnP algorithm). t [ s ] Pseudodeblended True
Deblended Fourier Difference Fourier Deblended PnP Difference PnP 1000 2000 x[m] 0.
t [ s ] 1000 2000 x[m] 1000 2000 x[m] 1000 2000 x[m] 1000 2000 x[m] 1000 2000 x[m]
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 ρ parameter.
For our 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.
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      30 Number of outer iterations 0. R e l a t i v e e rr o r (a) PnP stop training after 5 PnP stop training after 10 PnP stop training after      60 Number of outer iterations 0. R e l a t i v e e rr o r (b)
PnP, ε =1, epochs=10 PnP, ε =1, epochs=20 PnP, ε =1, epochs=30 PnP, ε =1, epochs=       30 Number of outer iterations 0.
R e l a t i v e e rr o r (c) PnP rerun with trained network PnP
288 a certain number of outer iterations leads to a stagnation in the error, or even worse to an increasein 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 goes down 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 [50].
Limitations and Conclusions
Limitations Our algorithm requires the setting of a number of hyperparameters, namely the numberof inner and outer iterations, the parameter ρ , 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.
Perhaps an adaptive strategy mayreduce the computational cost without impacting the quality of the reconstruction.
Since we warmstart each training process, one could hypothesize that some of the shallow features learned at oneiteration may be still valid for the subsequent denoising task.
Moreover, additional hyperparametersthat have not been explored in this work are associated with the network itself, e.g. the number oflayers, the activation function, batch size, etc.
This is also a direction for further research.
Anotherpossible drawback of our approach is the lack of convergence guarantees. PnP methods have beenshown to converge under certain assumptions, one of them being that the physical operator in the datamisfit term, in our case the blending matrix B , has to be strongly convex [38]. This requirement isnever met for the deblending problem, as B is always underdetermined.
Empirically, we observe thatx k and y k tend to converge to similar values for some carefully selected hyperparameters, indicatingthat at least in our experiments the algorithm converges successfully.
Moreover, the distance betweenx k and y k may be used as a suitable stopping criterion for the outer iterations, which is anotherhyperparameter that is currently set upfront.
Similarly, to obtain convergence guarantees for the PnPmethod, the denoiser has to be Lipschitz continuous; when a neural network is used, this means thatspectral normalization is required during training.
In [48], it was shown that PnP algorithms can beconvergent when combined 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 atshorter intervals, there is no difference compared to conventional acquisition.
Moreover, since alarger number of shots can be fired in the same overall acquisition time, recent research has suggestedthat the energy emitted by each source could be lowered.
This may provide acquisition solutions thatare 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 [28] 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 ρ 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. Checklist For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
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[Yes] We claim to introduce a new algorithm that combines
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deblending.
This is supported by our experiments.
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[Yes] We put an extensive paragraph inour Limitations and Conclusions section on the limitations of our approach.
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[Yes] We have performed a careful ablation study on a number ofparameters, and we provide references to the network architecture.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
[Yes] We provide an average for different seeds in the
supplementary material.
(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] We mention the machine we usedto run our experiments.
If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
[Yes] Yes, we acknowledgethe authors of the network we have used and cite their work and acknowledge the
publishers of the MobilAVO dataset.
(b) Did you mention the license of the assets?
[No] We cannot find the license of the
dataset.
We use MIT for the provided codes.
(c) Did you include any new assets either in the supplemental material or as a URL?
[No] We provide our codes in the supplementary material.
No new data or models are
provided.
(d) Did you discuss whether and how consent was obtained from people whose data you’re
using/curating?
[N/A]
We do not use data that involves people.
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content?
[N/A] We are not using data that contains personally
identifiable or offensive content