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1 Interpolation and Denoising of Seismic Data using Convolutional Neural Networks Sara Mandelli, Vincenzo Lipari, Paolo Bestagini, Member, IEEE, and Stefano Tubaro, Senior Member, IEEE Abstract—Seismic data processing algorithms greatly benefit from regularly sampled and reliable data. Therefore, inter- polation and denoising play a fundamental role as one of the starting steps of most seismic processing workflows. We exploit convolutional neural networks for the joint tasks of interpolation and random noise attenuation of 2D common shot gathers. Inspired by the great contributions achieved in image processing and computer vision, we investigate a particular architecture of convolutional neural network referred to as U-net, which implements a convolutional autoencoder able to describe the complex features of clean and regularly sampled data for reconstructing the corrupted ones. In training phase we exploit part of the data for tailoring the network to the specific tasks of interpolation, denoising and joint denoising/interpolation, while during the system deployment we are able to recover the remaining corrupted shot gathers in a computationally efficient procedure. We consider a plurality of data corruptions in our numerical experiments, including different noise models and different distributions of missing traces. Several examples on synthetic and field data illustrate the appealing features of the aforementioned strategy. Comparative examples show improvements with respect to recently proposed solutions for joint denoising and interpolation. I. I NTRODUCTION Seismic processing and imaging methods are essential to discover, localize and characterize economically worthwhile geological reservoirs, such as hydrocarbons accumulations, and to manage the extraction of the resources stored in them. However, since easy-to exploit resources are dramatically reducing and exploration targets are more and more complex, the requirements for the quality of seismic data, both in term of Signal-to-Noise ratio (S/N) and of regularity and density of its sampling, are constantly increasing. Unfortunately, various types of random and coherent noise, depending on the environment and on the acquisition technol- ogy, often corrupt seismic data sets. An additional problem is that economic limitations, cable feathering in marine case, environmental constraints and elimination of badly acquired traces cause irregular spatial sampling in almost all seismic acquisitions. Most state-of-the-art seismic processing algorithms, such as reverse-time-migration [1], full-waveform-inversion [2] and surface related multiple elimination [3] benefit from high quality regularly sampled data. Consequently, the vast majority of seismic processing workflows require data pre-processing steps, including effective denoising and trace interpolation S. Mandelli, V. Lipari, P. Bestagini and S. Tubaro are with the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano 20133, Italy (e-mail: [email protected]). algorithms. Moreover, due to the increasing size of the ac- quired data, a key factor of these procedures for industrial application is their computational burden, in terms of both memory requirements and computational time. The problems of trace interpolation and noise attenuation have been widely investigated, either simultaneously or sepa- rately. Among the dozens of interpolation methods have been proposed so far, we can roughly identify four main categories. Model-based algorithms implement an implicit migration- demigration pair [4], [5]. A major drawback of these tech- niques is that their performance is strongly affected in case of complex structural burden. A second approach, for both denoising and interpolation, is based on prediction filters [6], [7], [8], [9], which assume seismic data to be a (local) superposition of plane w1aves. However, these methods target regularly sampled data, which is a heavy limitation. Due to their repetitive features, clean seismic data are intrinsically low-rank in the time-space domain. Conversely, noise and missing traces increase the rank of the data [10]. Therefore, algorithms recasting the interpolation (and denois- ing) problems as rank reduction and matrix/tensor completion have been largely studied in the past decade as third alternative to the problem [11], [12], [13], [14], [15]. A great amount of denoising and interpolation algorithms exploit a transform domain where the clean signal can be represented only by few non-zero coefficients and therefore clean data and noise are more easily separable. The rationale behind this forth family of methods is that noise and missing traces map in non-sparse artifacts in the transform domain. Several fixed-basis sparsity-promoting transforms have been widely used also for seismic data interpolation. Among the various approaches, coming from different fields, we can cite: the Fourier transform [16], [17], the Hilbert-Huang transform [18], [19], the time-frequency peak filtering, [20], [21], [22], the Radon transform [23], different curvelet-like transforms [24], [25], [26], [27], [28], [29] and the EMD-seislet transform [30]. These methods assume the ability to describe the data in terms of a linear combination of atoms (i.e., the elemen- tary signals) taken from a dictionary (i.e., a predefined set of atoms). The aforementioned transform methods implicitly assume regularity of the data described by analytic models, resulting in predefined fixed dictionaries. However, these dic- tionaries can be thought as defining only a subset of the transforms methods. Alternatively, data driven sparse dictionaries can be learned directly from the dataset. In other words, these methods arXiv:1901.07927v4 [cs.NE] 21 Oct 2019

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Page 1: Interpolation and Denoising of Seismic Data using ... · a vast majority of seismic processing pipelines require data pre-processing steps, including effective denoising and trace

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Interpolation and Denoising of Seismic Datausing Convolutional Neural Networks

Sara Mandelli, Vincenzo Lipari, Paolo Bestagini, Member, IEEE, and Stefano Tubaro, Senior Member, IEEE

Abstract—Seismic data processing algorithms greatly benefitfrom regularly sampled and reliable data. Therefore, inter-polation and denoising play a fundamental role as one ofthe starting steps of most seismic processing workflows. Weexploit convolutional neural networks for the joint tasks ofinterpolation and random noise attenuation of 2D common shotgathers. Inspired by the great contributions achieved in imageprocessing and computer vision, we investigate a particulararchitecture of convolutional neural network referred to asU-net, which implements a convolutional autoencoder able todescribe the complex features of clean and regularly sampleddata for reconstructing the corrupted ones. In training phase weexploit part of the data for tailoring the network to the specifictasks of interpolation, denoising and joint denoising/interpolation,while during the system deployment we are able to recoverthe remaining corrupted shot gathers in a computationallyefficient procedure. We consider a plurality of data corruptionsin our numerical experiments, including different noise modelsand different distributions of missing traces. Several exampleson synthetic and field data illustrate the appealing featuresof the aforementioned strategy. Comparative examples showimprovements with respect to recently proposed solutions forjoint denoising and interpolation.

I. INTRODUCTION

Seismic processing and imaging methods are essential todiscover, localize and characterize economically worthwhilegeological reservoirs, such as hydrocarbons accumulations,and to manage the extraction of the resources stored in them.

However, since easy-to exploit resources are dramaticallyreducing and exploration targets are more and more complex,the requirements for the quality of seismic data, both in termof Signal-to-Noise ratio (S/N) and of regularity and densityof its sampling, are constantly increasing.

Unfortunately, various types of random and coherent noise,depending on the environment and on the acquisition technol-ogy, often corrupt seismic data sets. An additional problemis that economic limitations, cable feathering in marine case,environmental constraints and elimination of badly acquiredtraces cause irregular spatial sampling in almost all seismicacquisitions.

Most state-of-the-art seismic processing algorithms, suchas reverse-time-migration [1], full-waveform-inversion [2] andsurface related multiple elimination [3] benefit from highquality regularly sampled data. Consequently, the vast majorityof seismic processing workflows require data pre-processingsteps, including effective denoising and trace interpolation

S. Mandelli, V. Lipari, P. Bestagini and S. Tubaro are with the Dipartimentodi Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano20133, Italy (e-mail: [email protected]).

algorithms. Moreover, due to the increasing size of the ac-quired data, a key factor of these procedures for industrialapplication is their computational burden, in terms of bothmemory requirements and computational time.

The problems of trace interpolation and noise attenuationhave been widely investigated, either simultaneously or sepa-rately. Among the dozens of interpolation methods have beenproposed so far, we can roughly identify four main categories.

Model-based algorithms implement an implicit migration-demigration pair [4], [5]. A major drawback of these tech-niques is that their performance is strongly affected in case ofcomplex structural burden.

A second approach, for both denoising and interpolation,is based on prediction filters [6], [7], [8], [9], which assumeseismic data to be a (local) superposition of plane w1aves.However, these methods target regularly sampled data, whichis a heavy limitation.

Due to their repetitive features, clean seismic data areintrinsically low-rank in the time-space domain. Conversely,noise and missing traces increase the rank of the data [10].Therefore, algorithms recasting the interpolation (and denois-ing) problems as rank reduction and matrix/tensor completionhave been largely studied in the past decade as third alternativeto the problem [11], [12], [13], [14], [15].

A great amount of denoising and interpolation algorithmsexploit a transform domain where the clean signal can berepresented only by few non-zero coefficients and thereforeclean data and noise are more easily separable. The rationalebehind this forth family of methods is that noise and missingtraces map in non-sparse artifacts in the transform domain.

Several fixed-basis sparsity-promoting transforms have beenwidely used also for seismic data interpolation. Among thevarious approaches, coming from different fields, we can cite:the Fourier transform [16], [17], the Hilbert-Huang transform[18], [19], the time-frequency peak filtering, [20], [21], [22],the Radon transform [23], different curvelet-like transforms[24], [25], [26], [27], [28], [29] and the EMD-seislet transform[30].

These methods assume the ability to describe the data interms of a linear combination of atoms (i.e., the elemen-tary signals) taken from a dictionary (i.e., a predefined setof atoms). The aforementioned transform methods implicitlyassume regularity of the data described by analytic models,resulting in predefined fixed dictionaries. However, these dic-tionaries can be thought as defining only a subset of thetransforms methods.

Alternatively, data driven sparse dictionaries can be learneddirectly from the dataset. In other words, these methods

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assume that clean signals are a linear combination under asparsity constraint of the atoms in a learned overcompletedictionary (i.e., a frame made by a set of n functions withn > m, being m the signal dimensions). Learned dictionaries,in the form of explicit matrices for small patches, usually bettermatch the complex data characteristics. For instance, denoisingresults obtained using double sparse dictionary learning andoutperforming fixed dictionary transforms have been reportedin [31] and [32], combining the dictionary learning basedsparse transform with the fixed-basis transform, which iscalled double-sparsity dictionary. Recently, [33] introduced ajoint seismic data denoising and interpolation method using amasking strategy in the sparse representation of the dictionary.

In recent years, outstanding advancements brought by deeplearning and Convolutional Neural Networks (CNNs) havegreatly impacted whole signal and image processing commu-nity. In this context, innovative strategies for data interpolationand denoising based on deep learning have been proposedin many image processing tasks. Indeed, solutions based onCNNs are nowadays often exceeding state-of-the-art results.However, these methods have barely started to be explored bythe geophysical community for the problems of denoising andinterpolation. Promising results for the aforementioned taskshave been reported through residual neural networks [34],[35], generative adversarial networks [36] and convolutionalautoencoders [37], [38].

In this paper we leverage convolutional autoencoders bothto interpolate and to denoise irregular seismic data in the shot-gather domain. In particular, inspired by the important contri-butions achieved in image processing problems, we exploit aproperly trained U-net [39] as a strongly competitive strategyfor noise attenuation and reconstruction of missing traces inpre-stack seismic gathers. We provide examples on syntheticand field data showing promising performances on eitherdenoising, interpolation, or joint denoising and interpolationproblems. The results obtained for the joint interpolation anddenoising task also outperform a recent technique taken as areference.

In the following section section we introduce the proposednetwork architecture and the rationale behind the use of CNNsfor interpolation and denoising of 2D seismic data. This isfollowed by the specific workflow to be observed in order totrain and test the U-net for reconstructing corrupted gathers.Then, the proposed method is applied to synthetic and field2D seismic data. Finally, we discuss the advantages and thepotential issues of our method and draw conclusions.

We are aware that each research field has its specific jargon.Since the development of deep learning techniques applied toseismic data processing is still in its infancy, the terms used todefine the concepts related to CNNs might be not familiar toall the geophysical community. For this reason, we attach asappendix a short glossary in order to clarify some CNN-basedterminology which may be unusual for a geophysicist reader.

II. PROBLEM STATEMENT AND BACKGROUND ONAUTOENCODERS

In this section, we first report details about the formulationof the tackled problems, namely interpolation and/or denoising

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Fig. 1. Scheme of a Convolutional Autoencoder architecture.

of seismic data. Then, we provide some background conceptson Convolutional Autoencoders (CAs) and how to exploit themfor our specific goals, which is useful to understand the restof the paper.

A. Problem formulation

In this paper, we focus on the problem of reconstructingseismic gathers which have been corrupted by irregular tracesampling and/or additional noise. Formally, we represent eachoriginal non-corrupted seismic gather as I and its corruptedversion is denoted as I. Our goal is to estimate a clean anddense version of the seismic data, namely I, as similar aspossible to the original corresponding gather I.

In order to solve this problem, we make use of a particularkind of convolutional neural network named ConvolutionalAutoencoder (CA). Our choice is motivated by the greatcapability of CA in learning compact representations of thedata, and by the strong computational efficiency in recon-structing the corrupted ones. In the following, we reportsome backgrounds on CAs, introducing the specific networkarchitecture exploited for the prescribed task.

B. Convolutional Autoencoders for interpolation and denois-ing

Convolutional Autoencoders (CAs) are convolutional neuralnetworks whose architecture can be logically split into twoseparate components: the encoder and the decoder. The CAstructure is sketched in Fig. 1:• the encoder, represented by the operator E , maps the

input x into the so called hidden (or latent) representationh = E (x), which is the innermost encoding layer of theautoencoder, compressing the input x into a high-levelrepresentation [40];

• the decoder, represented by the operator D, transformsthe hidden representation into an estimate of the inputx = D(h).

For image processing problems, CA proves to be a verypowerful instrument for inpainting and denoising tasks [41],[42]. The rationale behind the use of CA for inpainting anddenoising shares some common concepts with the transform-based and dictionary learning techniques.

Indeed, CA is trained so that the encoder part results in acompact representation of clean data, where the interferencedue to noise and missing samples is not mapped. Therefore,if the compact representation is correctly built, the result ofthe decoder is a dense clean image without missing samples.

Page 3: Interpolation and Denoising of Seismic Data using ... · a vast majority of seismic processing pipelines require data pre-processing steps, including effective denoising and trace

3

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Concatenate

Conv2D + BN + Leaky ReLU

Conv2D + Leaky ReLU

Conv2D

ReLU + Conv2D! + Crop2D + BN + Dropout

ReLU + Conv2D! + Crop2D + BN

ReLU + Conv2D! + Crop2D

N⇥

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...

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N 16⇥

N 16<latexit sha1_base64="Z+NKVRwZs3FZMf+Pax+I3fgxRzQ=">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</latexit><latexit sha1_base64="Z+NKVRwZs3FZMf+Pax+I3fgxRzQ=">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</latexit><latexit sha1_base64="Z+NKVRwZs3FZMf+Pax+I3fgxRzQ=">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</latexit><latexit sha1_base64="Z+NKVRwZs3FZMf+Pax+I3fgxRzQ=">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</latexit>

N 2⇥

N 2<latexit sha1_base64="ljM2bqOvRFsZm9OXMBsqjsV4nB0=">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</latexit><latexit sha1_base64="ljM2bqOvRFsZm9OXMBsqjsV4nB0=">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</latexit><latexit sha1_base64="ljM2bqOvRFsZm9OXMBsqjsV4nB0=">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</latexit><latexit sha1_base64="ljM2bqOvRFsZm9OXMBsqjsV4nB0=">AAACXXicdVHBTtwwEPWmUGhKYWkPPfSA1VWrHtDKWcFuuCH10lNFpS4gbVYrx5ndWjiJZU8QyMqxX9MrfExP/ZU6IZWggnd6fm/G43lOtZIWGfvdC56trT/f2HwRvtx6tb3T3319asvKCJiKUpXmPOUWlCxgihIVnGsDPE8VnKUXnxv/7BKMlWXxHa81zHO+KuRSCo5eWvT3kqXhwn2t3aimCcocLL0vLfoDNhwfRZPxhLIha9GQaBzHMY06ZUA6nCx2ex+TrBRVDgUKxa2dRUzj3HGDUiiow6SyoLm44CuYeVpwP3Lu2k1q+qGyHEuqwVCpaCvC/Q7Hc2uv89RX5hx/2P+9RnzMm1W4jOdOFrpCKEQzCKWCdpAVRvqIgGbSACJvXg5UFlRwwxHBSMqF8GLlM3twaXYpte0WuLrbIEwyWPq/aE/OQLbQGmtnVmnt2HByODraZ/usDsPQJ/svPvo0OR0NI8+/HQyO4y7jTfKOvCefSEQm5Jh8ISdkSgT5SX6RG3Lb+xOsB1vB9l1p0Ot63pAHCN7+BTGXthA=</latexit>

N 4⇥

N 4<latexit sha1_base64="cVDJRwmKAftcmJCHndztngS5XB8=">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</latexit><latexit sha1_base64="cVDJRwmKAftcmJCHndztngS5XB8=">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</latexit><latexit sha1_base64="cVDJRwmKAftcmJCHndztngS5XB8=">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</latexit><latexit sha1_base64="cVDJRwmKAftcmJCHndztngS5XB8=">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</latexit>

N 8⇥

N 8<latexit sha1_base64="DNBdig5xZKg9Ht8Qh1vs5IWYGAE=">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</latexit><latexit sha1_base64="DNBdig5xZKg9Ht8Qh1vs5IWYGAE=">AAACXXicdVHBTtwwEPWmpdCUwtIeeuihVletekArB9HdcEPqhVMFUheQNquV48xuLZzEsieoyMqRr+EKH9NTfwUnBAkqeKfn92Y8nudUK2mRsb+94MXLlVera6/DN+tvNzb7W++ObVkZARNRqtKcptyCkgVMUKKCU22A56mCk/TsR+OfnIOxsix+4YWGWc6XhVxIwdFL8/6nZGG4cD9rF9c0QZmDpQ+leX/AhqO9aDwaUzZkLRoSjeI4plGnDEiHw/lW72uSlaLKoUChuLXTiGmcOW5QCgV1mFQWNBdnfAlTTwvuR85cu0lNv1SWY0k1GCoVbUV42OF4bu1FnvrKnONv+7/XiE950woX8czJQlcIhWgGoVTQDrLCSB8R0EwaQOTNy4HKggpuOCIYSbkQXqx8Zo8uzc6ltt0Cf+42CJMMFv4v2pMzkM21xtqZZVo7Nhx/39nbZtusDsPQJ3sfH32eHO8MI8+Pdgf7cZfxGvlIPpNvJCJjsk8OyCGZEEEuyRW5Jje9f8FKsB5s3JUGva7nPXmE4MMtSE22HA==</latexit><latexit sha1_base64="DNBdig5xZKg9Ht8Qh1vs5IWYGAE=">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</latexit><latexit sha1_base64="DNBdig5xZKg9Ht8Qh1vs5IWYGAE=">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</latexit>

2⇥

2<latexit sha1_base64="OOSo0bxgVacI41NOWVhRWVIEqPo=">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</latexit><latexit sha1_base64="OOSo0bxgVacI41NOWVhRWVIEqPo=">AAACRnicdVBNb9QwEJ0sXyV8tXDkYrECcagiZ1V201slLhyLxLYVu6uV48wuVh3HsicVVZR/wRV+D3+BP8ENccVJg0QRzOn5vRm/mZdbrTxx/i0a3bh56/adnbvxvfsPHj7a3Xt84qvaSZzLSlfuLBcetTI4J0Uaz6xDUeYaT/Pz151+eoHOq8q8o0uLq1JsjdooKShQ7ydsSapEzybr3TFPpofpbDpjPOF9dSCdZlnG0oEZw1DH673oxbKoZF2iIamF94uUW1o1wpGSGtt4WXu0Qp6LLS4CNCL4rJp+5ZY9r72gill0TGnWk/jnRCNK7y/LPHSWgj74v7WO/Je2qGmTrRplbE1oZGdESmNv5KVTIQtkhXJIJLrNkSnDpHCCCJ1iQspA1iGca58WF8r64YCPVxfEywI3IfT+1Tgs1tZS27ht3jY8mb2aHO7zfd7GcRyS/R0f+z84mSQpT9K3B+OjbMh4B57CM3gJKczgCN7AMcxBgoFP8Bm+RF+j79GP6OdV6ygaZp7AtRrBL5S8r34=</latexit><latexit sha1_base64="OOSo0bxgVacI41NOWVhRWVIEqPo=">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</latexit><latexit sha1_base64="OOSo0bxgVacI41NOWVhRWVIEqPo=">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</latexit>

N⇥

N<latexit sha1_base64="vOiNWvH72jTVa+o+dx2byysTTWo=">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</latexit><latexit sha1_base64="vOiNWvH72jTVa+o+dx2byysTTWo=">AAACRnicdVBNb9QwEJ0sXyV8tfTIxWIF4lCtnKrsprdKXHqqisS2FburlePMLlYdx7InFVWUf8EVfg9/gT/BDfVaJw0SRTCn5/dm/GZeZrXyxPmPaHDn7r37DzYexo8eP3n6bHPr+YkvKydxKktdurNMeNTK4JQUaTyzDkWRaTzNzt+1+ukFOq9K84EuLS4KsTZqpaSgQH08YnNSBXp2tNwc8tF4P5mMJ4yPeFctSMZpmrKkZ4bQ1/FyK3o9z0tZFWhIauH9LOGWFrVwpKTGJp5XHq2Q52KNswCNCD6Lulu5Ya8qL6hkFh1TmnUk/jlRi8L7yyILnYWgT/5vrSX/pc0qWqWLWhlbERrZGpHS2Bl56VTIAlmuHBKJdnNkyjApnCBCp5iQMpBVCOfWp/mFsr4/4PPNBfE8x1UIvXvVDvOltdTUbp01NR9N3u7u7/Ad3sRxHJL9HR/7PzjZHSUBv98bHqR9xhvwAl7CG0hgAgdwCMcwBQkGvsBX+BZ9j35Gv6Krm9ZB1M9sw60awDX9Mq+2</latexit><latexit sha1_base64="vOiNWvH72jTVa+o+dx2byysTTWo=">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</latexit><latexit sha1_base64="vOiNWvH72jTVa+o+dx2byysTTWo=">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</latexit>

N16⇥

N16

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N2⇥

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N4⇥

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N8⇥

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Fig. 2. Architecture of the used U-net.

Consequently, it is possible to train a CA to learn a hiddenrepresentation of the clean data in common shot gathers andthen to recover clean and densely sampled gathers from noisyand scattered ones.

In particular, in this work we exploit a CNN architectureknown as U-net. Originally designed for image segmentationproblems and then used for several different tasks [43], theU-net is named after the shape it is usually graphicallyrepresented with. Indeed, U-net shares a large part of thearchitecture with classical CA. However, in a U-net, therepresentations of the input obtained at different levels ofthe encoder are directly concatenated to the correspondingdecoder levels. This peculiarity typically allows to preserve thestructural integrity of the image and to enable a very precisereconstruction. For the sake of brevity, we refer the interestedreaders to [39] for a detailed explanation of these architectures.

Continuing the analogy with the transform-based methods,we can think the trained U-net as an instrument implicitlyproviding a multi-scale/multi-resolution compact representa-tion, able to describe the complex features of clean seismicdata where noise and missing data are not modeled. By using acomputer vision terminology, we can think at the interpolationand denoising task as an image transfer problem, with thegoal of transforming gathers corrupted by noise and/or missingtraces into regularly sampled clean gathers.

III. RECONSTRUCTING THE CORRUPTED GATHERS

In this section we explain how to reconstruct the corruptedseismic gathers. In particular, we report the technical detailsconcerning the used network architecture and then we describetwo strategies for network training, to be used according tothe specific data corruption. Finally, we describe the systemdeployment, that is the procedure designed to process, afterthe training stage, the actual gather we want to reconstruct.

A. Implementation of U-net

In order to focus on local portions of the gathers and toensure a sufficiently large amount of data under analysis, wework in a patch-wise fashion. Specifically, we divide eachshot gather into K patches of size N ×N . Then, we alwaysconsider a single patch as input to the network. Notice that,for what regards the U-net implementation, patches can beextracted from the gather with or without overlap, being the U-net architecture independent from the patch selection strategy.The CNN we propose takes inspiration by recent works forthe interpolation task [38] but introduces a few novelties inorder to simplify the network architecture at highest levels andenhance the system efficiency without drops in performances.For instance, as seismic data typically has a value range verydifferent from that of natural digital images for which CNNsare typically studied, each patch is multiplied by a constantgain G (i.e., G = 2000 in the following experiments), whichproves to be an effective data normalization procedure interms of convergence speed and achieved validation results.Moreover, we do not need to consider the gradient computationand the batch normalization after the first convolutional layer,thus making the proposed approach leaner.

Considering that corrupted gathers are labeled as I, thegeneric k-th corrupted patch given as input to the networkis denoted as Pk.

As U-net like architectures turn out to be the state-of-the-art for the tasks of image inpainting [44], [45] and denoisingof medical images [46], we follow the trend started by [39],exploiting a U-net architecture composed by the blocks shownin Fig. 2:

1) A number of stages where a 2D Convolution with filtersize 4×4 and stride 2×2, sometimes followed by BatchNormalization (BN) and/or Leaky ReLU, is performed.These stages lead to the hidden representation (i.e., theresult of the encoding part). Specifically, we do not

Page 4: Interpolation and Denoising of Seismic Data using ... · a vast majority of seismic processing pipelines require data pre-processing steps, including effective denoising and trace

4

include the BN stage in the outermost layer and afterthe 2D Convolution leading to h. It is worth noting thatthe number of filters increases from 64 to 512 as we godeep into the network.

2) The same number of stages as before where a ReLU,a 2D Transposed Convolution with filter size 4 × 4 andstride 2 × 2 and a 2D cropping, possibly followed byBN and Dropout are performed. In this case, we use BNon all layers except from the last one, while Dropout isused only in the initial layers, until we reach a patchdimension of N

16 × N16 . In each stage we concatenate the

result of the corresponding encoding stage as in a typicalU-net fashion. Note that the number of filters is graduallydiminished as we go up in the right path of the network(i.e., decoding path). The last stage outputs the patch Pk,of the same size of the input patch.

Additionally, the overall architecture scales according to thepatch dimension N , which can be selected depending on pos-sible application-driven constraints. Anyway, as the networkcan be characterized by more than 40 million parameters, itneeds to be trained on a significant amount of seismic imagesas any typical deep learning solutions.

B. U-net trainingEach corrupted patch Pk coming from training dataset DT

is processed by the U-net, and a patch Pk is estimated. Thenetwork weights w are estimated by minimizing the lossfunction computed with the original uncorrupted patch Pk.For the task of interpolation only, some samples from Pk arealready known, therefore we include their knowledge in theloss function.

First, a set of K patches is extracted from image I; eachpatch Pk is processed by the U-net architecture U(·,w), anda patch Pk is estimated. For the tasks of denoising only andjoint denoising/interpolation, the post-processing block simplydivides Pk by the gain G. Concerning the task of interpolationonly, also the known samples of Pk are used together withPk for reconstructing the patch Pk. Finally, image I isreconstructed by re-assembling together the estimated patches.Whether the patch extraction process required some patch-overlap, the overlapping portions are sample-wise averaged.

Once defined the network architecture, the key point is thedesign of the training strategy through a proper definition ofthe cost function tailored to our specific problem. Indeed, theU-net defines a parametric model x = U(x,w), between theoutput x and the input x and network weights w. The trainingphase consists in estimating the network weights w throughthe minimization of a distance metrics between the networkinput and its output. This distance is usually referred as lossfunction, and its minimization is carried out using iterativetechniques (e.g., stochastic gradient methods, etc.).

Specifically, as shown in Fig. 3, we train the network in or-der to transform patches Pk, extracted from gathers corruptedby noise and/or missing traces, into regularly sampled andclean patches Pk. As in any supervised learning problem, weassume to have a training dataset DT and a validation datasetDV , each one composed by pairs of corrupted/uncorruptedgathers (I, I).

Interpolation only

Fig. 3. Sketch of the training phase.

These datasets are exploited for estimating the networkparameters w and to decide when to stop the cost functionminimization process. Actually, the training stage slightlydiffers depending of the specific problem: denoising only,interpolation only, and joint denoising/interpolation.

For denoising only and joint denoising/interpolation, modelweights are estimated by minimizing the loss function betweenPk and Pk, defined as the squared error over all patchesbelonging to gathers in the training set DT . Formally,

w = arg minw

∑Pk∈DT

∥∥∥Pk − Pk(w)∥∥∥2F, (1)

where ‖·‖F represents the Frobenius norm and, with a slightabuse of notation, Pk ∈ DT denotes patches extracted fromgathers belonging to training dataset.

However, for the task of interpolation only, we a-prioriknow that only some samples need to be reconstructed bythe network (i.e., the missing ones), whereas the others canbe left untouched. For this reason, the loss is evaluated onlyon reconstructed samples. This is implicitly performed byadding a masking stage that sets to zero all uncorrupted traces.Network weights are then estimated as

w = arg minw

∑Pk∈DT

∥∥∥(Pk − Pk(w))⊗M∥∥∥2F, (2)

where ⊗ represents the Hadamard product and M is a binarymask of size N ×N defined as

[M]i,j =

{1 if

[Pk

]i,j

is a missing sample

0 otherwise.(3)

As in standard neural network training, we follow aniterative procedure to minimize either (1) or (2), stopping atthe iteration where the mean squared error over all the patchesPk ∈ DV is minimum.

Specifically, we use Adam optimization algorithm [47], withlearning rate and patience (i.e., the number of epochs with noimprovement after which training will be stopped) initializedat 0.01 and 10, respectively. The former is decimated while thelatter is halved in presence of plateau of the cost function. Ingeneral, we train the network for a maximum number of 100epochs, although we verified the smallest loss on validationpatches is often achieved within the first 30 training epochs.

C. System Deployment

When a new corrupted gather I belonging to evaluationset DE is under analysis, its recovered version is estimatedfollowing the scheme depicted in Fig. 4.

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Patch

Extraction

Image

Reconstruction

Post

Processing

Interpolation only

Fig. 4. Proposed procedure for recovering each corrupted gather I.

First of all, a set of K patches is extracted from image Ias described in the previous Section. Then, each patch Pk isprocessed by the U-net architecture in order to estimate thepatch Pk.

Following the same logic of the training phase, the esti-mated patches are post-processed in slightly different waysaccording to the specific goal. For denoising only and jointdenoising/interpolation, each patch Pk simply undergoes adenormalization step, thus it is divided by the gain G to obtainthe output patch Pk. Concerning the task of interpolation only,it is reasonable to leave the known samples untouched in thefinal estimated patch. Therefore, exploiting the binary maskdefined in (3), each patch Pk is obtained as

Pk =Pk ⊗ M + Pk ⊗M

G, (4)

being M the the logical complement (i.e., the negation) of M.Eventually, in order to reconstruct the image gather I, all the

estimated patches Pk are re-assembled together, sample-wiseaveraging the overlapping portions if some overlap betweenpatches was used during patch extraction procedure.

IV. RESULTS

In this section we present the result of our experiments,obtained on well-known public datasets. Specifically, we eval-uate our methodology over both synthetic and field data. First,we introduce the accuracy metrics we exploit for evaluatingthe results, and we show a tutorial example considering a verysimple case of study. Second, we separately validate the perfor-mances of the proposed interpolation and denoising strategieson both synthetic and field data. Finally, we investigate thecombined interpolation and denoising problem also comparingour method against a recent solution.

A. Accuracy metrics

We evaluate the performances of our method in reconstruct-ing each entire seismic image belonging to the evaluation setDE , namely the corrupted gathers which have never been seenby the U-net. The accuracy metrics is the same used by [35]and [36], i.e., the S/N, defined as the ratio between the powerof the original gather and the power of the reconstruction error:

S/N = 10 log10

|| I ||2F|| I− I ||2F

I ∈ DE . (5)

B. Tutorial example

In order to show the effectiveness of our strategy, we startwith a very simple case of study, that is, the corrupted gatherunder investigation consists of a single dipping event with aRicker wavelet. In order to train, validate, and test the network,we build a dataset of 251 gathers with size 512×256, randomlyvarying slope and depth of the event. In particular, we alwaystrain and validate the network using the first 250 gathers, andwe test over the last one. From each gather, we extract 153patches with size 128 × 128 and with stride (24, 16) in thetemporal and spatial domain, respectively. For what concernsthe rationale behind the patch selection methodology, we referthe reader to the next section, where we explain in details themotivations behind our choice.

We consider three different data corruptions: (i) randomlylocated missing traces; (ii) additive white Gaussian noise(AWGN); (iii) joint presence of noise and missing traces.Concerning the first situation, we randomly delete the 30% ofthe traces from each gather; in the second case, we add AWGNnoise with standard deviation equal to 0.1; the third exampleconsiders both corruptions. Results of the gather reconstructionare shown in Fig. 5. The achieved S/N are 34.2dB, 19.5dB and20.1dB, respectively. It is worth noting that, in all the threesituations, we are able to reconstruct the corrupted shot gatherwith an acceptable level of accuracy, introducing few signalleakage.

To compare our results with more standard solutions, weconsider an algorithm based on shaping regularization forrandom missing traces recovery [17] and a method based on f-x deconvolution for random noise attenuation [9]. Concerningthe interpolation of missing traces, the S/N achieved by [17] is33.5dB, thus it is comparable with our result. Regarding thedenoising problem, the method based on f-x- deconvolutiononly achieves S/N = −2.3dB. Despite the big gap betweenthis solution and ours, notice that the initial S/N of the gathercorrupted by AWGN was −15.8dB, therefore also the f-xdeconvolution strategy is able to reconstruct the gather with anacceptable accuracy. Furthermore, contrarily to our proposalwhich is data-driven, f-x deconvolution is totally blind anddoes not need training data for estimating the result.

These examples show that a properly trained U-net is able todeal with the tasks of interpolation and denoising: the networkis able to recognize the features characterizing linear events,and to effectively reject as noise all other components. Asa matter of fact, the training set has been designed in orderto incorporate prior knowledge that actual signals are linearevents. However, the main benefit of U-net and more generallyof all CNNs is that, through learning, they are in principle able

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(a)

(c)

(b)

(d)

Fig. 5. Tutorial example results: (a) shows the original gather; (b) depicts results for interpolation of random missing traces; (c) draws results for AWGNdenoising; (d) shows results for joint interpolation and denoising. For each corruption, first column reports the corrupted gather, second column the reconstructedone. For problems (b) and (d), third column shows the reconstruction error (i.e., the difference between reconstructed and original), while for problem (c) itreports the residual error (i.e., the difference between reconstructed and corrupted).

to handle complex situations which cannot be easily describedby standard approaches based on mathematical models ofnoise and seismic events.

C. Problem 1: Interpolation of missing tracesIn this section we present the network performances in

interpolating gathers with missing traces. In particular, we startshowing results achieved over a synthetic dataset, consideringvarious interpolation situations. Finally, we evaluate the pro-posed strategy on real field data.

1) Synthetic data set: The reference dataset used to sys-tematically explore the results is extracted from the wellknown synthetic BP-2004 benchmark [48]. In particular, wework with 1348 shot gathers, cropped at the first 1152 traces(taking the source as reference) and at the first 1920 timesamples/trace. The central frequency of each trace is 27Hz,sampled every δt = 6ms, and the group spacing is 12.5m.

In order to properly evaluate the proposed method, we ran-domly split the dataset into training, validation and evaluation,

using 250 shot gathers for training and validation (further spliton 75% of images for training set DT , and 25% for validationset DV ), and the remaining for evaluation set DE .

a) Interpolation of uniformly distributed missing traces:The first experiment investigates the situation of randomlylocated missing traces with uniform distribution. This choicefollows the main reasoning of the works proposed in literature[38], [17], [33].

In order to simulate a seismic acquisitions with randomlymissing traces, we extract 3 different datasets from the refer-ence one, deleting a percentage H of the available data traces.To be precise, for each shot gather I, we randomly delete theH% of its traces, H ∈ {10, 30, 50}, obtaining a holed gatherI.

As shown in Section III-A, we work in patch-wise fashionfor reconstructing the corrupted gathers. Specifically, eachgather entering the network is initially split into a pluralityof squared patches, with dimensions N ×N . In light of this,we perform an initial experiment to analyze the behaviour of

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Fig. 6. Average S/N achieved on gathers belonging to DE , as a function ofpatch dimension N and overlap.

network output as a function of the specific input data. Thegoal of this primary investigation is to select a good patchextraction method, that is, the strategy leading to the highestreconstruction accuracy on the evaluation set. We considerdifferent values for N , namely N ∈ {16, 32, 64, 128, 256},and we evaluate the cases of non-overlapping patches and ofpatches extracted with an overlap of N/2 in both directions.

To evaluate the U-net performances according to the chosenpatch extraction method, we use S/N defined in (5). Fig. 6shows the average S/N achieved over gathers belonging toevaluation set, with and without the overlap between theextracted patches. Note that the case N = 256 does notinclude results without overlap because the gather dimensionsare not integer multiples of this value. It is noticeable thatsmall values of N are not good solutions for reconstructing thecorrupted images, probably because the U-net needs to analyzemore samples together in order to find a significant hiddenrepresentation of the input patch. As expected, introducingsome overlap during patch extraction always returns betterperformances than just selecting adjacent patches. This is dueto two main factors: first, selecting overlapped patches in-creases the amount of data seen by the network and reasonablyimproves its performances; second, in the image reconstructionphase, the overlapping portions of the patches are sample-wise averaged, decreasing the possibility to generate undesirededge/border effects.

Even if selecting an overlap of N/2 gives slightly betterresults, one consideration must be done. At training stage,we found out that a good strategy is to group in a singlebatch all the patches extracted from the same shot gather,ending up with a batch size (i.e., the amount of patches ina batch) strictly dependent on N . Notice that the number ofsamples per batch does not change with N , if patches arenot overlapped. Conversely, in case of overlapped patches,the number of samples per batch increases, as some samplesbelong to multiple patches. Therefore, the higher the overlap,the larger the amount of GPU memory required in trainingphase. If the absence of overlap requires a GPU memory usagemore or less equal to 4GB for every N , in case of overlap therequired space increases in a quadratic fashion.

(a)

(b)

(c)

(d)

Fig. 7. Example of data interpolation considering one gather of the syntheticdataset. (a) depicts the original gather I, cropped in its central portion with size450×300; (b) reports the corrupted gather I, with 50% of randomly missingtraces, (c) shows the reconstructed gather I; (d) depicts the reconstructionerror, which is the difference between reconstructed and original shot gather.

Therefore, considering that the achieved S/N performancesof the two methodologies (overlapped and non-overlappedpatches) are not so far, we choose the patch extraction strategywhich selects only adjacent and non-overlapping patches.For this reason, hereinafter we only investigate the networkbehavior considering non-overlapping patches, as overlappingpatches would make the solution impractical in the majorityof cases.

Regarding the patch dimension N , as the S/N curvemonotonically increases with the patch dimension but withoutdropping performances in terms of memory usage, we selectN = 128 for all the experiments. We end up with batches of135 non overlapping patches with dimensions 128 × 128 ex-tracted from each shot gather. The process involves more than25 000 training patches, more than 8 500 validation patches,and more than 145 000 testing patches for each dataset.

Regarding the results, we are able to achieve S/N of32.8dB, 24.2dB and 18.8dB for H = 10, 30 and 50, re-spectively. The processed gathers do not visually show anyartifacts due to the interpolation method, even with H = 50,as depicted by the example in Fig. 7. For what concerns the

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Fig. 8. Markov chain of the burst corruption model.

required computational time, we need more or less 75 minutesfor training and validate the network over 100 epochs. Despitethis could seem a quite huge amount of time, once traininghas been done, we are able to recover a corrupted shot gatherbelonging to the test set in less than 0.3 seconds. Specifically,we run our tests on a Workstation equipped with Intel XeonE5-2687W v4 (48 Cores @ 3 GHz), RAM 252 GB and 1TITAN V (5120 CUDA Cores @ 1455MHz), 12 GB.

b) Interpolation of bursts of missing traces: Uniformdistribution of missing traces, described by the percentage H ,allows the evaluation of average reconstruction performancesof the interpolation. However, in order to have a more detaileddescription, here we study the performances of the proposedU-net-based interpolation on a more sophisticated corruptionmodel. Basing on the consideration that missing traces (due forinstance to spatial obstacles) are likely to appear in groups, wepropose a bursty missing traces model inspired by the packetloss models of telecommunication networks [49]. The termburst is taken from the field of telecommunication networks,and refers to groups of consecutive events (in this case, groupsof consecutive missing traces). The more bursty a distributionof missing traces is, the more the missing traces are likely tocluster.

In particular, the model is a two states Markov model de-scribed by two parameters, α and β: α refers to the probabilityof a missing trace, while β is the average length of the burst,i.e., the average number of missing traces which are adjacentone another.

The Markov chain of the model is depicted in Fig. 8, whereNM represents the non-missing trace state whereas M is thestate for missing trace. The probability to find a corruptedtrace, given that the previous one (in the spatial dimension)was missing, is q, while the probability to pass from a nonmissing trace to a missing one is p. These probability valuescan be derived from α and β, formally,

q = 1− 1

β, p =

α

β(1− α). (6)

Exploiting this model, we can simulate more realistic sce-narios, where clusters of adjacent missing traces can occur,due for instance to environmental constraints or sudden inter-ruptions during acquisitions. In order to test our method onthis missing trace distribution, we select various percentageof missing traces α ∈ {10, 30, 50}% with average burstlength β ∈ {1, 2, 3}, corresponding to 12.5m, 25m and 37.5mrespectively. Notice that, the larger the average gap, the greaterthe gap size dispersion. Indeed, for β = 1 the standarddeviation is equal to 0 traces (isolated missing traces only); onthe contrary, for β = 2 and β = 3 the standard deviations are

TABLE IAVERAGE S/N [DB] ACHIEVED ON GATHERS BELONGING TO DE , FOR

DIFFERENT VALUES OF α AND β .

α β 1 2 3

10 38.4 25.3 21.9

30 32.8 21.7 18.1

50 29.9 18.7 15.7

σ = 1.14 and σ = 2.44 traces, respectively. For instance, inthe datasets under examination, β = 3 provides a maximumgap up to 30 traces (corresponding to 375m), which simulatesa quite large physical obstacle.

Table I depicts the average results achieved by the U-net onthe evaluation set. Notice that, the larger the burst length, thelower the resulting S/N. This enlightens the need of furtherinvestigations for interpolating bursts of many traces: as amatter of fact, as the size of the group of adjacent missingtraces increases, the ability of the network in reconstructing theunknown samples diminishes. Nonetheless, notice that evenin the worst case, i.e., (α, β) = (50, 3), the U-net is able tomaintain acceptable reconstruction performances.

c) Interpolation by transfer learning: In order to test therobustness of the proposed method in interpolating missingdata, we generate two further synthetic datasets, exploitingthe very same acquisition geometry and model of the datasetpreviously presented, but with different sampling rates 4msand 8ms. The goal of this experiment is to check if the U-netarchitecture, when trained on data sampled every 6ms, is ableto reconstruct differently-sampled data. This is an exampleof the well-known transfer learning strategy [40]. Namely, itcorresponds to analyzing the performances of one networkwhich has already been trained over a dataset having differentfeatures from the testing one.

To this purpose, we propose to select as test case the uniformmissing traces framework, randomly deleting the 30% of tracesfrom these new datasets. Then, we evaluate the reconstructionresults on gathers belonging to the evaluation set DE ofthese datasets, with the difference that we exploit the networktrained on the dataset sampled every δt = 6ms.

Average results of the interpolation are shown in Table II.Notice that we report also the interpolation results we canachieve if following the standard training procedure, that is,training the network using data with the same sampling time ofthe evaluation set. Even if the difference between the resultsis noticeable, Fig. 9 shows an example of δt = 8ms datareconstruction exploiting the U-net trained on δt = 6ms. It isnoticeable that the error is concentrated in the upper part ofthe gather and in the areas with many adjacent missing traces,while in the other regions the reconstruction is quite good andacceptable.

d) Interpolation of regularly missing traces: The lastcase we investigate for the synthetic dataset is that of recov-ering regularly missing traces, which can be considered asupsampling the acquisition geometry and has several practicalimplications, e.g., interpolation in the cross-line direction. Thiscase has its own peculiarities, and it is often the most challeng-

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(a)

(b)

(c)

(d)

Fig. 9. Example of transfer learning data interpolation. (a) depicts the originalgather I, sampled every 8ms and cropped in its central portion with size450×300; (b) shows the corrupted gather, with 30% of randomly distributedmissing traces; (c) reports the reconstructed gather I, exploiting the U-nettrained on data sampled every 6ms; (d) shows the reconstruction error, i.e.,(c) - (a).

TABLE IIAVERAGE S/N [DB] ACHIEVED ON DE , FOR SAMPLING TIME δt = 4MS

AND δt = 8MS.

δt 4ms 8ms

Train on 6ms 10.1 10

Train on δt 25.3 23.8

ing interpolation problem, especially when the dips are aliased.As a matter of fact, methods based on sparse transformationsand low-rank constraints have limited application in this casebecause of the strong spatial aliased energies. We want to testthe conjecture that network training extracts some high-levelcharacteristics of seismic data which are more robust to aliasthan linear events or low rank assumptions, therefore implicitlyincluding an anti-aliasing strategy.

We consider two scenarios. The first one consists in trainingthe U-net with new data, generated by regularly deleting the50% of the shot gathers traces, and then following the sameprocedure previously shown for the shot gather reconstruction.

The second solution emulates the transfer learning techniquepresented above: we use the network trained on data with 50%randomly missing traces to reconstruct the shot gathers with50% regularly missing traces. The achieved S/N are 30dBand 22.8dB for the first and second solution, respectively.Results achieved over a shot gather region characterized bysteep dips are shown in Fig. 10. Notice that, by training andtesting the U-net with the same data, the reconstruction erroris very low, while with the transfer learning technique theerror contains again some signal content. Nonetheless, thisis an expected results and both errors seem acceptable atvisual inspection. Indeed, we select as a reference a recentindustrial software based on f-x deconvolution which does notrequire training data (hence can be considered more or lessanalogous to the transfer learning technique), and it achievesS/N = 22.7dB on the same data. For the sake of clarity, Fig. 11includes the absolute value of the Fourier spectrum computedfor the original shot gather, for the corrupted one, and for theestimated one by means of the first proposed strategy. It isworth noting that the alias introduced in the corrupted shotgather can be deleted by our reconstruction method.

2) Field data: In this section, we apply the U-net forreconstructing corrupted real seismic data and compare theresult achieved with those obtained in [38]. To this purpose,we exploit as field data the well known Mobil Avo VikingGraeben Line 12 dataset [50]. Specifically, this dataset consistsof 1001 marine shot gathers. Each gather is composed of 128traces of 1408 time samples, with temporal sampling of 4msand receiver sampling of 25m.

In order to compare our results with those obtained by[38], we simulate a seismic acquisition with a uniform dis-tribution of randomly missing traces. Therefore, for eachacquired gather I, we randomly delete the H% of its traces,H ∈ {10, 30, 50}, obtaining a scattered sampled gather I.

Following the same rationale of the synthetic example, wesplit each dataset into 250 gathers for training and validationand leave the remaining to evaluation set. Then, in order toachieve a similar number of patches per gather (i.e., 135 in thesynthetic case), we extract 129 patches with size 128 × 128,overlapped only on the temporal dimension, specifically withpatch-stride of 10 samples. Notice that, in this case, thepresence of patch overlap does not cause issues in memoryusage, as the number of samples entering the network issimilar to the chosen configuration for the synthetic example.

Results obtained on the evaluation dataset DE are reportedin Table III, while Fig. 12 shows an example of gatherreconstruction where 50% of traces is missing. It is noticeablethe improvement in performances of the proposed architecture,This achievement is due to the specific changes performedon the U-net architecture as described in Section III-A. Asa matter of fact, even with a reduced amount of gathersfor training and validation (i.e., 25% of the whole datasetinstead of 75%), the resulting S/N always exceeds the pastperformances.

In order to build a preliminary evaluation of the resultswhich would be obtained by seismic imaging algorithms ondata reconstructed by U-net, we migrate the sections of theoriginal Viking Graben dataset, of the corrupted dataset with

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(a) (b) (c)

(d) (e) (f)

Fig. 10. Interpolation of regularly missing traces. (a) depicts the original gather, cropped around samples (1100 – 1400, 100 – 300); (b) shows the corruptedgather which contains half the original traces; (c) depicts the reconstructed gather by training the U-net on data with regularly missing traces; (d) reports thereconstructed gather by transfer learning; (e) shows the reconstruction error of the standard strategy, i.e., (c) - (a); (f) is the reconstruction error of the transferlearning approach, i.e., (d) - (a).

(a) (b) (c)

Fig. 11. Absolute value of Fourier spectrum of shot gathers for the problemof interpolation of regularly missing traces. (a) depicts the spectrum of theoriginal gather cropped around samples (1100 – 1400, 100 – 300); (b) showsthe spectrum of the corrupted gather which contains half the original traces;(c) depicts the spectrum of the reconstructed gather by training the U-net ondata with regularly missing traces.

TABLE IIIAVERAGE S/N [DB] ACHIEVED ON GATHERS FROM DATASET IN [50].

H 10 30 50

Proposed U-net 25.7 20.5 16.7

[38] 22.5 15.1 10.2

50% randomly missing traces and of the dataset reconstructedby the proposed methodology. As a comparison, we migratealso the dataset reconstructed using the nonlinear shapingregularization method proposed by [17]. The images depictedin Fig. 13 show that, in case of data reconstructed with U-net,noise is attenuated and loss of continuity is almost perfectlyrecovered in the migrated section. This occurs also for thereconstruction of [17], however, contrarily to that of U-net,the migrated section shows some spurious events which are

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(a)

(d)

(c)

(b)

Fig. 12. Example of data interpolation, considering one gather of the fielddataset [50]. (a) depicts the original gather I; (b) reports the corrupted gatherI, with 50% of randomly missing traces; (c) shows the reconstructed gatherI; (d) reports the reconstruction error, i.e., (c) - (a).

(a) (b)

(c) (d)

Fig. 13. Kirchhoff migrated section of the Viking Graben dataset: (a);migration of original data; (b) migration of corrupted data; (c) migration ofdata reconstructed by U-net; (d) migration of data reconstructed by the methodproposed in [17].

not present in the original image.

D. Problem 2: Denoising of corrupted gathers

In this section we report the results of the numericalexperiments related to denoising of corrupted gathers. Weconsider two kinds of additive noise, the standard additivewhite Gaussian noise (AWGN), and a spike-like noise.

Concerning the patch extraction methodology to be applied,we select exactly the same strategy of that presented in SectionIV-C1. As a matter of fact, the presence of randomly missingtraces as well as the additive random noise can be seen as twogeneric kinds of gather corruption, which can be tackled bythe U-net in a similar way.

1) AWGN noise model: In order to test our method over aplurality of signal to noise ratios (S/N), we add white gaussiannoise for achieving S/N = S ∈ {−3, 0, 3}dB, defined as theratio between the signal and noise power.

The average results obtained on shot gathers belonging tothe evaluation set DE are the following: S/N = 12.8dB,14.4dB and 16.3dB, correspondent to increasing values ofS. Indeed, these results can be considered an upper boundfor the achievable performances of the proposed strategy inrealistic scenarios. As a matter of fact, they are obtained onthe assumption to have clean data available for the trainingphase, which is never the case for field acquired data.

2) Spike-like noise model: Pre-stack seismic data can beaffected by different types of random noise coming from var-ious sources, such as wind motion, poorly planted geophonesor electrical noise, most of these being far more complex thansimple AWGN. For instance, some of these seismic noisesexhibit spike-like characteristics [51] and are lately gaininggrowing interest, as they strongly affect the processing ofsimultaneous source data acquired from recent seismic surveys[52].

Therefore, we propose to use our network for denoising datacorrupted by additive spike-like noise. In order to simulate thisnoise, we add spiky noise with variable density d%, namely thepercentage of corrupted samples in one gather. In particular,the binary values of this noise are set to the minimum andmaximum values of the original uncorrupted data. Then, weconvolve each noise trace with a Ricker wavelet having thesame central frequency of the data (i.e., 27Hz) and unit energy.This way, we generate two corrupted datasets, correspondingto d ∈ {1, 3}.

Fig. 14 shows an example of spike-like corruption denoisingfor d = 3. It is noticeable that, even if the corrupted imagevisually undergoes a strong degradation, the reconstructed onepresents almost all the features of the original data. Moreover,Fig. 14(d) reports the residual error between corrupted andreconstructed gather. It is worth noting that a large portion ofthe useful signal remains unaltered, except for very reducedareas. This trend is confirmed by Table IV, which reports theaverage results achieved on set DE .

3) Towards standard denoiser emulation: In order to high-light the U-net-based method versatility, we propose to exploitour denoising strategy as an emulator of some well knownstandard denoising algorithms. In particular, we select two

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(b)

(c)

(a)

(d)

Fig. 14. Example of spike-like noise attenuation. (a) reports the originalgather; (b) reports the corrupted gather, with noise density d = 3; (c) showsthe reconstructed gather; (c) shows the residual error, i.e., (b) - (c).

TABLE IVINITIAL S/N [DB] ON CORRUPTED GATHERS AND FINAL S/N [DB]

ACHIEVED BY SPIKE-LIKE DENOISING.

d 1 3

Initial −34.8 −39.6

Final 16.8 12.3

noise attenuation strategies, namely a Wavelet based denoising[53] and a f-x deconvolution method included in a recentproduction software. To test the denoising performances, weuse the datasets corrupted by AWGN with S/N = S ∈{−3, 0, 3}dB.

Initially, we process the whole datasets through the afore-mentioned standard denoising algorithms. Through this oper-ation, we are generating denoised data which can be more orless considered in the same way as clean uncorrupted gathers.

In a second phase, we train our network in a slightlydifferent way than the approach shown Section III-B. Indeed,we train the U-net substituting to ground truth gathers thoseobtained through denoising by Wavelet or by the industrialf-x deconvolution. Thus, the training step includes pairs of

TABLE VAVERAGE S/N [DB] ON GATHERS BELONGING TO DE , TRAINING THEU-net ON GATHERS DENOISED BY WAVELET (LEFT TABLE) AND BY THE

INDUSTRIAL SOFTWARE BASED ON F-X DECONVOLUTION (RIGHT TABLE).

S −3 0 3

U-net 4.8 5.7 6.9

Wavelet 4.7 5.7 6.9

S −3 0 3

U-net 7 8.6 10.3

f-x dec. 7 8.7 10.3

noisy gathers and gathers denoised by standard algorithms.Eventually, we evaluate denoising results on shot gathersbelonging to DE , comparing reconstructed gathers with theoriginal ones, as described in (5). Specifically, we compute theaverage results for U-net and for standard denoising algorithmsas well.

From results depicted in Table V it is quite evident thatperformances of U-net are comparable with those achievedthrough the denoising algorithm used for training, showingthat U-net is able to mimic their performances.

Moreover, the proposed method has a further advantage,which is the low computational effort in denoising a genericgather. As a matter of fact, if a limited amount of timeis needed for training the network model parameters, theevaluation phase is very efficient: the optimized productionsolution and the Wavelet denoising take respectively the sametime and 25% more than the time required by our strategy(i.e., approximately 0.3 seconds) for estimating each denoisedgather.

These results pave the way for one potential application ofour method in realistic situations. Indeed, as previously stated,having clean gathers available for the training phase is notthe case when dealing with real data. Furthermore, denoisingfield acquisitions often require complex and computationallyexpensive algorithms.

In order to overcome these issues, we recommend ourstrategy as a viable alternative to many standard denoisingalgorithms. Specifically, when a large field dataset is available,the following algorithm can be applied:

1) randomly select a subset of the acquired shot gathers;2) perform an accurate and computationally expensive de-

noising on the selected shot gathers;3) train the U-net on the selected pairs of acquired/denoised

gathers;4) make use of the trained U-net to denoise the remaining

data.After a certain dimension of the dataset, due to the fixedcomputational cost for denoising the selected subset and train-ing the U-net, the application of the U-net-based denoisingbecomes computationally cheaper than denoising the wholedataset with the standard noise attenuation algorithm. Indeed,the computational advantage of U-net increases with thedimension of the dataset and the complexity of the denoisingalgorithm.

E. Complete problem: Joint interpolation and denoisingThe last situation we propose is the more realistic case of

study, implying additive noise corruption jointly with missing

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TABLE VIAVERAGE S/N [DB] ACHIEVED ON GATHERS BELONGING TO DE , FOR

EACH DATASET EXTRACTED FROM BP-2004 [48].

H S −3 0 3

10 12.2 13.8 15.6

30 11.5 12.9 14.4

50 10.4 11.6 12.9

traces. We investigate two cases of study: the former exploitsthe same synthetic dataset of all the previous experiments,while the latter uses a different dataset, comparing our methodwith state-of-the-art techniques.

1) AWGN and uniform missing traces: In order to in-vestigate if the proposed method is able to retrieve theoriginal synthetic data, we consider the presence of AWGNand uniformly distributed missing traces. Likewise previouslydone, we add noise leading into S ∈ {3, 0,−3}dB anddelete a percentage H ∈ {10, 30, 50} of the available datatraces for simulating seismic acquisition with irregular receiversampling and missing traces. This way, we generate 9 differentdatasets, corresponding to various combinations of additivenoise and missing traces. Table VI resumes the average resultsobtained on shot gathers belonging to the evaluation set DE ,considering all possible combinations of missing traces andadditive noise variances.

2) Comparison with a recent data-driven method: To com-pare our strategy with a recent learning-based algorithm,we consider the Double-Sparsity Dictionary Learning methodproposed by [33] and one strategy based on fixed dictionarytransform used as baseline in [33], i.e., the Curvelet method.In order to perform a fair comparison, we reproduce exactlythe same synthetic example provided in [33].

Specifically, the dataset is extracted from the BP-1997benchmark [54] and includes 385 shot gathers, consideringthe last 240 receivers (taking the source as reference) with384 samples/trace. We add noise and delete some traces inthe dataset following the procedure described therein. In thefirst stage, as done in [33], we normalize the range of eachtrace to 1. Then, white gaussian noise is generated, low pass-filtered with a cut-off frequency of 30Hz and finally addedto the traces. We perform the same numerical experimentsproposed in [33], testing plenty of noise standard deviationsσ ∈ {0.05, 0.10, 0.15, 0.20, 0.25} and missing traces’ percent-ages H ∈ {10, 20, 30, 33, 40, 50, 60}.

For what concerns the training phase, we randomly select250 shot gathers for training and validation (split in 75% −25%). In test phase, we process exactly the same image ofthat used in [33], namely the first shot gather in the dataset.Note that, in order to be fair, we never use this shot gather inthe training phase.

During training, we extract from each image 153 overlap-ping patches with size 128 × 128 and stride (16, 14) alongrows and columns, respectively. This operation has been donein order to achieve training and validation sets with similar size(concerning the number of patches) to the previously shownsituations. Notice that we consider exactly the same training-

validation-evaluation procedure of that depicted in SectionsIII-B and III-A.

For comparing the results, we use the evaluation metricsproposed in [33], namely the peak signal to noise ratio(PS/N), defined as

PS/N = 10 log10

smax

σ2(I− I), (7)

being smax the dynamic range of the clean signal, thus fixedto 1.

Fig. 15(a) shows the original shot gather without noiseadded and missing traces (i.e., the ground truth of the ex-periment). The corrupted version of the gather with 33% ofmissing traces and σ = 0.1 is shown in Fig. 15(b). Figs. 15(c)-(d) show the recovered gathers obtained with double-sparsitydictionary learning and U-net, respectively. We can noticethat there are some events which are well reconstructed bythe U-net while are missing in the retrieved shot gather viadouble-sparsity dictionary learning. Specifically, Figs 15(e)-(f) show the error panels (i.e., the difference between therecovered images (c) and (d) and the ground truth (a)) for theresults obtained with the state-of-the-art technique and U-netrespectively. It is quite evident that the error correspondingto double-sparsity dictionary learning is more affected byresidual coherent events, meaning that those are not correctlyrecovered. These qualitative considerations are confirmed bythe corresponding PS/N values: 32.1 dB for double-sparsitydictionary learning and 33.7 dB for U-net. Clearly, for bothmethods the reconstruction error is not optimum and there isstill room for improvement. Nonetheless, it is worth noting thatthe limited size of the shot gathers (only 240×384, versus the1152 × 1920 of BP-2004 benchmark and the 128 × 1408 ofMobil Avo Viking Graeben Line 12 dataset) could potentiallyundermine the ability of U-net to learn how to describethe complex features of the clean data without modelingnoise and missing data. Indeed, to be trained, CNNs usuallyneed a substantial amount of data for achieving acceptableperformances. Whether more data samples were accessible,we expect the performances to improve accordingly.

Fig. 16 displays the performances of different reconstruc-tion methods by varying the missing traces ratio and selectingσ = 0.1. In particular, we compare results reported in [33]with our results, averaged over 100 different realizations ofthe column pattern used for randomly deleting the traces. It isnoticeable that we significantly outperform both the dictionarylearning-based method and the Curvelet-based, gaining anaverage of 2.4 dB with respect to the former strategy and6.1 dB to the latter one.

Fig. 17 reports the achieved results for a plurality ofnoise standard deviations. The performances of the U-net aresignificantly superior than those of dictionary learning-basedstrategy, in all the examined cases. Moreover, our method re-veals to be more robust in presence of strong noise. As a matterof fact, as noise standard deviation σ increases, the curvesrelated to state-of-the-art method decay in a worse fashionthan ours, to the point that we can achieve PS/N = 30.3dB for H = 50 and σ = 0.25, against the 26.7 dB of thedictionary learning-based technique.

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(a) (c)

(b)

(e)

(d) (f)

Fig. 15. Shot gather with 33% of missing traces and σ = 0.1. (a) shows the original gather; (b) depicts the corrupted version; (c) shows the recovered gatherobtained with double-sparsity dictionary learning, PS/N = 32.1 dB; (d) shows the recovered gather obtained with U-net, PS/N = 33.7 dB; (e) illustratesthe reconstruction error of double-sparsity dictionary learning, i.e., (c) - (a); (f) depicts the reconstruction error of our method, i.e., (d) - (a).

Fig. 16. Results of different reconstruction methods by varying the missingtraces ratio H and for σ = 0.1.

Fig. 17. Results of U-net and double-sparsity dictionary learning reconstruc-tion methods by varying the missing traces ratio H = {10, 30, 50} and forσ = 0.05 : 0.05 : 0.25.

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V. CONCLUSIONS

We proposed a method for reconstruction of corruptedseismic data, focusing on noise attenuation and interpolationof missing pre-stack data traces in the shot-gather domain. Inparticular, we considered random noise cases with differentstatistics and a variety of missing traces distributions. Ourapproach makes use of a convolutional neural network ar-chitecture for interpolation and denoising of 2D shot gathers,showing performance improvements, in terms of S/N ratio,with respect to recent solutions for joint denoising and inter-polation.

Results achieved on controlled synthetic experimentsdemonstrate that the proposed method is a promising strategyfor seismic data pre-processing. The method is capable ofeffectively and efficiently restore 2D corrupted data, and itis also able to deal with the task of spatially upsampling theshot gathers. Moreover, once the network training procedureis completed, processing data with our strategy is also quiteefficient in terms of computational effort.

We examined also the potential application of our methodol-ogy on field data for production environments. In this situation,it is interesting to notice that the proposed algorithm canbe used also to emulate the effect of more time consumingclassical data pre-processing strategies.

Future work will be devoted to investigating issues related todenoising of field data, exploring for instance the feasibility ofa transfer learning procedure by training convolutional neuralnetworks on properly designed synthetic data and testing onfield data. Moreover, investigations are needed for denoisingof more challenging types of coherent noise and artifactsaffecting seismic data (e.g., ground roll in land data) anddifferent kinds of missing data (e.g., missing short offsets andcross-line upsampling).

Further studies on the network architectures and loss func-tions (e.g., considering the use of variational autoencoders,generative adversarial networks, etc.) could relax the need ofcorrupted/uncorrupted pairs of gathers for the training set, thushelping in dealing with the problem of building a trainingdataset for denoising.

In addition, investigations are needed to carefully examinethe effect of the proposed strategy on standard processing andimaging algorithms (e.g., surface related multiple elimination,reverse time migration, etc.).

Finally, other aspects we would like to examine for movingtowards production environments are exploiting the similaritybetween adjacent gathers to improve the reconstruction per-formances and the extension of the proposed procedure to 3Ddata.

In the light of the promising results achieved with theproposed architecture of convolutional neural network, webelieve this tool can pave the way for even more efficientand accurate solutions.

VI. GLOSSARY ON CNNS

Deep learning and CNNs are still relatively new topics forthe geophysical community. In order to ease the readabilityof the paper, this glossary reports the definition of some

standard CNN-related terms which have been used throughoutthe paper. Since our experiments have been conducted usingKeras framework, for a thorough reading, please refer to [55].

• Activation function: it is a scalar function (typicallynonlinear) that is pointwise applied to the output of alayer.

• Batch: the set of data (e.g., images or image-patches)used for one iteration (i.e., a gradient update step) duringmodel training.

• Batch Normalization (BN): this layer applies a transfor-mation to the data in order to maintain the mean close to0 and the standard deviation close to 1, for each batch.

• Concatenate: layer that concatenates a list of inputs,which have all the same dimensions except for theconcatenation axis.

• Convolutional layer: it is the core layer of CNN archi-tecture. The convolutional layer consists of nodes thatperform a convolution between an input and a set offilters (convolutional kernels). In the 2D case, the inputimage (or image-patch) to an arbitrary convolutionallayer can be defined as Iin(x, y, c), with c ∈ [1, ..., Nc].Specifically, x and y are the 2D image dimensions andc the so-called “channel” dimension. Each convolutionallayer has a set of Nj bidimensional filters: a generic filteris defined as wj(x, y, c). The output of the layer will be:

Iout(x, y, j) = J

(Bj(x, y)+

Nc∑c=1

Iin(x, y, c) ∗wj(x, y, c)

)∀j ∈ [1, ..., Nj ], (8)

where Bj(x, y) is a bias matrix, ∗ is the symbol for the2D convolution on x and y dimensions, and J(·) is anactivation function. The filter weights are set during thelearning stage. Notice that the input size Nc depends onthe number of filters on the previous convolutional layer.

• Dropout: this layer randomly sets to zero the update ofsome network node at each iteration during the trainingphase. This can help prevent overfitting. In our experi-ments, Dropout layer sets to zero the 50% of neurons.

• Epoch: one pass, during network training, over the entiretraining dataset. Since the CNN model updates every timea single batch of data is processed, the model is updatedmultiple times (i.e., the number of batches) during oneepoch.

• Leaky ReLU: it is a leaky version of a Rectified LinearUnit. Leaky ReLU is used to avoid too many zerosflooding the network; each negative signal sample x isconverted to α · x, with α typically small (e.g., α = 0.2in our experiments):

LeakyReLU(x) =

{x if x > 0

α · x if x ≤ 0.

• Learning rate: this parameter controls how much to adjustthe CNN weights with respect to the loss gradient. Thelower the learning rate, the slower we descend towardsthe minimum of loss function. The corresponding concept

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in seismic inverse problem terminology is that of updatestep-size.

• Loss: it is the cost function which measures how far thenetwork predictions are from the desired results. This isthe function to be minimized during the network training.

• Patience: the number of epochs with no improvementin the validation loss function (or validation accuracymetrics) after which the training will be stopped.

• ReLU: the Rectified Linear Unit is an activation functionwhich converts to 0 the negative signal samples, leavinguntouched the positive ones:

ReLU(x) = max (0, x) .

• Stride: in a convolutional layer, applying a stride k in theith dimension means that convolutions are computed bymoving the kernel with a step of k samples along the ith

dimension. The output of a convolution with stride k inthe ith dimension is equivalent to downsampling at ratek along the ith dimension of the output of a standardconvolution.

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