GNGTS 2021 - Atti del 39° Convegno Nazionale

GNGTS 2021 S essione 3.1 396 G ( x S , x R ) is defined by a N ×N binary matrix M ( x S , x R ) (i.e., a mask) as M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise 0 acquired with the actual acquisition geometry G can be modeled as the to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We consider the umber of sources is smaller than the number of sources in the ideal dense m X 0 , we can invert for X the equation X 0 = M ⨀ X , which is an ill-posed this problem through the deep image prior paradigm (Ulyanov et al., 2018), re itself can capture enough low-level features (i.e., our prior) from a single ume, mainly exploiting the self-similarities in the data. Through this paradigm, arametric function ^ X = F θ ( Z ) mapping a noise realization Z to the data space. y initialized and updated by minimizing a loss function on the known traces, in en the generated data ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 only on the actual acquisition geometry, the output of the CNN ranges over the twork learns how to transform a fixed noise volume Z into the decimated data elf; the missing traces are restored at the same time. ET WITH SYMMETRIC OUTPUT nneberger et al., 2015), which has been designed for medical image processing, as a deep prior for seismic shot gathers interpolation (Kong et al., 2020; Liu et odified architecture based the rationale that seismic data exhibit self-similarity at the output of the network (i.e. the reconstructed time slice) should be s of the modified architecture are the following: e features of seismic data at different scales we use the multi-resolution block. convolutional layers extracts features at different scales, and their outputs are r. Moreover, a residual connection with 1×1 convolution, which proved to be ation problems, is added; in the sequence is gradually increased, in order to efficiently manage memory; ection, distinctive feature of standard U-Net, is replaced by the path residual s the 3×3 and 1×1 convolutions. Its non-linear transformations on encoder semantic gap introduced by deeper decoder stages; rformed by 3×3 convolutions with stride 2×2, upsampling is performed by s instead of deconvolutions, improved the results; Therefore, the dataset follows: M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise Therefore, the dataset X 0 acquired with the actual acquisition geometry G can be modele application of the mask M to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We con case in which the actual number of sources is smaller than the number of sources in the ide dataset. To recover X starting from X 0 , we can invert for X the equation X 0 = M ⨀ X , which is an i inverse problem. We solve this problem throug the deep image prior paradigm (Ulyanov et al. where the CNN architecture itself can capture enough low-level features (i.e., our prior) from corrupted s ismic data volume, mainly expl iting the self-similarities in the data. Through this p the CNN is modeled as a parametric function ^ X = F θ ( Z ) mapping a noise realization Z to the dat The weights θ are randomly initialized and updated by minimizing a loss function on the known t this case the distance between the generated data ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the loss is computed only on the actual acquisition geometry, the output of the CNN ranges whole ideal survey: the network learns how to transform a fixed noise volume Z into the decim from the decimated data itself; the missing traces are restored at the same time. M ULTI - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-net architecture (Ronneberger et al., 2015), which has been designed for medical image pr has been successfully used as a deep prior for seismic shot gathers interpolation (Kong et al., 202 al., 2019). Here, we use a modified architecture based the rationale that seismic data exhibit self-s at different scales and that the output of the network (i.e. the reconstructed time slice) s symmetric. The main points of the modified architectur are the following: • in order to capture the features of seismic data at different scales we use the multi-resolutio A sequence of 3×3 convolutional layers extracts features at different scales, and their ou concatenated together. Moreover, a residual connection with 1×1 convolution, which prov ffective for interpolation problems, is added; • the number of filters in the sequence is gradually increased, in order to efficiently manage • the direct skip-connection, distinctive feature of standard U-Net, is replaced by the path block which includes the 3×3 and 1×1 convolutions. Its non-linear transformations on features ba ance the semantic gap introduced by deeper decoder stages; • downsampling is performed by 3×3 convolutions with stride 2×2, upsampling is perfo bilinear interpolation instead of deconvolutions, improved the results; ac ired with the actual acquisition ge metry follows: M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwis Therefor , th dataset X 0 acquired with the actual cquisition geometry G can be modeled as t application of the mask M to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We consider t case in which the ac ual nu ber of sources is smaller than the number of sources in the ideal den dataset. To recover X starting from X 0 , we can invert for X the equation X 0 = M ⨀ X , which is an ill-pos inverse problem. We solv this problem throug the deep image prior paradigm (Ulyanov et al., 2018 where the CNN architecture itself can capture enough low-level features (i.e., our prior) from a sing corrupted seismic data volume, mainly exp oiting the self-similaritie in the da a. Throug this paradig the CNN is modeled as a parametric function ^ X = F θ ( Z ) mapping a noise realization Z to the data spac The weights θ are randomly initialized and updated by minimizing a loss function on the known traces, this case the dist nce betw en the generated d ta ^ X nd the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the loss is computed only on the actual cquisition geometry, the ou put of the CNN ranges over t whole ideal survey: the etwork learns how to transform a fixed noise volume Z into the decimat d d from the decimat d data itsel ; the missing trac s are res ore at the same time. M ULTI - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-net archi ecture (Ronneberger et al., 2015), which has been designed for medical image processin has be n successfully used as a deep prior fo seismic shot gathers int polation (Ko g et al., 2020; Liu al., 2019). He e, we use a modifi d architecture based th r tionale that seismic data exhibit self-similari at ifferent scales and that the output of the network (i.e. the reconstructed time slice) should symmetric. The main points of the modified ar hitecture are the following: • in order to capture the f atures of s ismic data at different scales we use the multi-resolution bloc A quen e of 3×3 convolutional layers extracts fe tures at different scales, and their outputs a concat nated tog ther. Moreover, a residual con ectio with 1×1 convolution, which proved to ffective for interpolation problems, is added; • the number of filters in the sequence is gradually increased, in order to efficiently manage memor • the direct skip-conne tio , distinct ve feature of standard U-Net, is replaced by the pa resid block which includes the 3×3 and 1×1 convolutions. Its non-linear transformations on enco features balance the s mantic gap introduced by deeper decoder stages; • downsampling is performed by 3×3 convolutions with str de 2×2, upsampling is performed bilinear int rpolations instead of dec nvolutions, improved the results; can be model d as the application of the mask follows: M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise Therefore, the dataset X 0 acquired with the actual acquisition geometry G can be mo application of the mask M to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We case in which the actual number of sources is smaller than the number of sources in the dataset. To recover X starting from X 0 , we can invert for X the equation X 0 = M ⨀ X , which is inverse problem. We solve this problem through the deep image prior paradigm (Ulyanov where the CNN architecture itself can capture enough low-level features (i.e., our prior) f corrupted seismic data volume, mainly exploiting the self-similarities in the data. Through th the CNN is modeled as a parametric function ^ X = F θ ( Z ) mapping a noise realization Z to th The weights θ are randomly initialized and updated by minimizing a loss function on the kno this case the distance between the generated data ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the loss is computed only on the actual acquisition geometry, the output of the CNN ra whole ideal survey: the network learns how to transform a fixed noise volume Z into the d from the decimated data itself; the missing traces are restored at the same time. M ULTI - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-net architecture (Ronneberger et al., 2015), which has been designed for medical imag has been successfully used as a deep prior for seismic shot gathers interpolation (Kong et al., al., 2019). Here, we use a modified architecture based the rationale that seismic data exhibit s at different scales and that the output of the network (i.e. the reconstructed time slic symmetric. The main points of the modified architecture are the following: • in order to capture the features of seismic data at different scales we use the multi-res A sequence of 3×3 convolutional layers extracts features at different scales, and thei concatenated together. Moreover, a residual connection with 1×1 convolution, which effective for interpolation problems, is added; • the number of filters in the sequence is gradually increased, in order to efficiently mana • the direct skip-connection, distinctive feature of standard U-Net, is replaced by the block which includes the 3×3 and 1×1 convolutions. Its non-linear transformation features balance the semantic gap introduced by deeper decoder stages; • downsampling is performed by 3×3 convolutions with stride 2×2, upsampling is bilinear interpolations instead of deconvolutions, improved the results; t the ideal survey follows: M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise Therefore, the dataset X 0 acquired with the actual acquisit on geometry G can be mod ap lication of the mask M to the ideal surv y X for ach time slice, i.e., X 0 = M X . We case in which the actual number of sources is smaller than the number of sources in the dataset. To recover X starting from X 0 , we can invert for X the equation X 0 = M ⨀ X , which is inverse problem. We solve this problem through the de p image prior paradigm (Ulyanov et where the CNN architecture itself can capture enough low-level features (i.e., our prior) fr corrupted seismic data volume, mainly exploit ng the self-similarities in the data. Through thi the CNN is modeled as a parametric function ^ X = F θ ( Z ) map ing a noise realization Z to the The weights θ are randomly initialized and updated by minimizing a los function on the kno this case the distance betwe n the generated data ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the los is computed only on the actual acquisition geometry, the output of the CNN ran whole ideal survey: the network learns how to transform a fixed noise volume Z into the de from the decimated data itself; the missing traces are restored at he same time. M ULTI - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-net architecture (Ron eberger et al., 2015), which has been designed for medical image has been suc es fully used as a de p prior for seismic shot gathers interpolation (Kong et al., al., 2019). Here, we use a modif ed architecture based the rationale that seismic data exhibit se at dif erent scales and that the output of the network (i.e. the reconstructed time slice) sym etric. The main points of the modified architecture are the following: • in order to capture the features of seismic data at different scales we use the multi-resol A sequence of 3×3 convolutional layers extracts features at different scales, and their concatenated together. Moreover, a residual connection with 1×1 convolution, which effective for interpolation problems, is ad ed; • the number of filters in the sequence is gradually increased, in order to efficiently manag • the direct skip-connection, distinctive feature of standard U-Net, is replaced by the p block which includes the 3×3 and 1×1 convolutions. Its non-linear transformations features balance the semantic gap introduced by de per decoder stages; • downsampling is performed by 3×3 convolutions with stride 2×2, upsampling is p bilinear interpolations instead of deconvolutions, improved the results; r a time slice, i.e., foll ws: M ( x S , R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise Therefore, the dataset X 0 acquired with e actual acquisition geometry G can be model application of the mask M to he ideal survey X for each time slice, i. ., X 0 = M ⨀ X . We co case in whic the actual number of sources is smaller than the number of sources in the id dataset. To recover X starting from X 0 , we can invert for X the quation X 0 = M ⨀ X , whic is an inverse problem. We solve this problem through the d ep image prior par digm (Ulyanov et a wher the CNN architecture itself can capture enough low-level f atures (i.e., our prior) fro corrupted seismic data volume, mainly exploiting the self-similar ties in the data. Through t is the CNN is modeled as a parametric function ^ X = F θ ( Z ) mapping a noise realization Z to the d The weights θ are randomly in tialized and updated by min miz ng a loss function the known this case the distance between the generated data ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the loss i computed only on the actual acquisition geometry, the output of the CNN rang whole ideal survey: the network learns how to transform a fixed noise volume Z into the deci from the decimated ata itself; the missing traces are restored at the same time. M ULTI - RESOLUTION U-N ET WITH SYM ETRIC OUTPUT The U-net architecture (Ronneberger et al., 2015), which has been designed for medical image p has been successfully used as a deep rior for seismic shot gathers interpolation (Kong et al., 20 al., 2019). Here, we use a modified architecture based the rationale that seismic data exhibit self at different scales and that the output of the network (i.e. the r constructed time slice) sym etric. The main points of the modified architecture are the foll wing: • in order to capture the features of seismic data at different scales we use the multi-resolut A sequence of 3×3 convolutional layers extracts features at differ nt scales, and their o concatenated together. Moreover, a residual connection with 1×1 convolution, which pr effective for interpolation problems, i added; • the number of filters in the sequenc is gradually increased, in orde to effic ently manage • the direct skip-connection, distinctive f ature of standar U-Net, is replaced by the pat block which includes the 3×3 and 1×1 convolutions. Its non-linear transformations o features balance the semantic gap introduced by deeper decoder stages; • ownsampling is performed by 3×3 convolutions with stride 2×2, upsampling is perf bilinear interpolations instead of deconvolutions, improved the results; . We consider the case in which the actual number of sources is smaller than the number of sources in the ideal dense dataset. To recover etry G ( x S , x R ) is defined by a N ×N binary matrix M ( x S , x R ) (i.e., a mask) as M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise X 0 acquired with the actual acquisition geometry G can b modeled as the k M to th ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We consid r the ual number of sources is smaller than the number of sources in the ideal dense from X 0 , w can invert for X the equation X 0 = M ⨀ X , which is an ill-posed olve this problem through the deep image prior paradigm (Ulyanov et al., 2018), ecture itself can capture enough low-level features (i.e., o r prior) from a single volume, mainly exploiting the self-similar t es in the data. Through this paradigm, s a param tric function ^ X = F θ ( Z ) mapping a noise realiza ion Z to the dat space. omly initialized and updated by m nimizing a loss function on the known traces, in etween the generated data ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 uted only on the actual acquisition geometry, the output of the CNN ranges over th ne work learns how to transform a fixed noise volume Z into the decimated data ta itself; the missing traces are restored at the same time. U-N ET WITH SYMMETRIC OUTPUT (Ronneberger et al., 2015), which has been designed for medical image proces ing, used as a deep prio for eismic sho g thers interpolation (Kong et al., 2020; Liu et e a modified archit cture based the rationale that se smi data exhibit self-similarity d that the output of the n twork (i.e. the reconstructed time slice) should be oints of the modified architecture are the following: re the features of seismic data at diff ent scales we use the multi-resolution block. ×3 convolutional layers extra s features at different scales, and their outputs are ether. Moreover, a residual con ection with 1×1 convolution, which proved to be rpolation problems, is added; lters in the sequence is gradually increased, in order to efficiently manage memory; onnection, distinctive featur of standard U-Net, s replaced by the path residual ludes the 3×3 and 1×1 convolutions. Its non-linear transformations on encoder the semantic gap introduced by deeper decoder stages; performed by 3×3 convolutions wi stride 2×2, upsampling is performed by tions instead of deconvolutions, improved the results; starting from Any acquisition geometry G ( x S , x R ) is defined by a N ×N binary matrix M ( x S , x R ) (i. follows: M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise Therefore, the d taset X 0 acquired with the actual acquisition geometry G can be m application of he mask M to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . W case in which the actual number of sources is smaller than the number of sources in th dataset. To recover X starting from X 0 , we can invert for X the equation X 0 = M ⨀ X , which i inverse problem. We solve this problem through the deep image prior paradigm (Ulyanov where the CNN architecture itself can capture enough low-level features (i.e., our prior) f corrupted s ismic data volume, mainly exploiting the self-similarities in the data. Through t the CNN is modeled as a par metric function ^ X = F θ ( Z ) mapping a noise realization Z to t The weights θ are randomly initialized and updated by minimizing a loss function on the kn t s case the distance betwee the gen rated d ta ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the loss is computed only on the actual acquisition geometry, the output of the CNN r whole ideal survey: the network learns how to transform a fixed noise volume Z into the d from the decimated data itself; the missing traces are restored at the same time. M ULTI - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-net architecture (Ronneberger et al., 2015), which has been designed for medical ima has been successfully used as a deep prior for seismic shot gathers interpolation (Kong et al. al., 2019). Here, we use a modified architecture based the rationale that seismic data exhibit at different scales and that the utput of the network (i.e. the reco stru t d time slic symmetric. The main points of the modified archite ture are the following: • in order to capture the features of seismic data at different scales we use the multi-res A sequence of 3×3 convolutional layers extracts features at different scales, and the concatenated together. Moreover, a residual connection with 1×1 convolution, which effective for interpolation problems, is added; • th n mber of filters in the sequence is gradually increas , in order to efficiently man • the direct ski -connection, distin tive feature of standard U-Net, is replaced by the block which i clu es the 3×3 and 1×1 convolutions. Its non-linear transformation features balance the emantic gap introduced by deeper decoder stages; • downsampling is performed by 3×3 convolutions with stride 2×2, upsampling is bilinear interpolations instead of deconvolutions, improved the results; , we can invert for Any acquisition geometry G ( x S , x R ) is defined by a N ×N binary follows: M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise Therefore, the dataset X 0 acquired w h the actual acquisition ge application of the mask M to the ideal survey X for each time slice, i. case in which the actual number of sources is smaller than the num dataset. To recover X st rting from X 0 , we can invert for X the equation X inverse problem. We s lve t is proble through the deep image prio where the CNN archi ecture itself can capture enough low-level f a corrupted seismic data volume, ma nly explo ting the self-similarities i the CNN is modeled as a parametric function ^ X = F θ ( Z ) mapping a no The we ghts θ are randomly initialized and updated by m nimizing a lo this case the dist n between the enerat d data ^ X and original dat θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the l ss is computed only the actual acquisition geometry, the whole ideal su vey: the network learns how t transform a fixed noise from the decimated dat itself; the missing traces are restored at the sa M ULTI - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-net architecture (Ronneberger et al., 2015), which has been desi has been successfully used as a deep prior for seismic shot gathers inte al., 2019). Here, we use a modified a chitecture based the rationale that at different scales and h t the output of the network (i.e. r symmetric. The ma n points of the m dified arch te ture are the follow • in order to capture the features of seismic data at diff rent scal s A sequence of 3×3 convolutional layers ext ac s fe tures t dif concatenat d t gether. Moreover, a residual nection wit 1× effective for interp lation problems, is added; • he numb r of f lte s in the sequenc i gradually increased, in or • the direct skip-c nnection, distinctive feature of standard U-N block which includes the 3×3 and 1×1 co volu io . Its non- features b lance the semantic gap intr uced by deeper decoder s • d wnsampling is performed by 3×3 convolutions with stride b lin r interpolations instead of deconvolution , improved the re the equation Any acquisition geometr G ( x S , x R ) s defined by a N ×N b nary atrix M ( x S , x R ) (i.e., a mask) follows: M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise Therefore, th dataset X 0 a quired with the actual acquisition g ometry G can be modeled as th application of the mask M to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We o sider th case in which th actual number of sources is smalle than the number of sources in the ideal dens dataset. To recover X starting from X 0 , we can invert for X the equation X 0 = M ⨀ X , which is an ill-pose inverse problem. We solve this proble through the deep image prior paradigm (Ulyanov et al., 2018 where the CNN architecture itself can cap enough low-level features (i. ., our prior) from a singl corrupted seismic data volume, mainly exploiting the self-similarit es in the data. Through this paradig the CNN is modeled as a parametric f nction ^ X = F θ ( Z ) mapping a no se realization Z to the data spac The weights θ are rando ly in tialized and updated by minimizi g a loss functio on the known traces, i this case th distance between the generated data ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the loss is computed only on the actual acquisition geometry, the output of th CNN ranges over t wh le ideal survey: the etwork learns how to transform a fixed noise volume Z into the decimated da from the decimated dat itself; the missing traces are restored at the same time. M ULT - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-n t architectu (Ronn berger et al., 2015), which has been designed for medical image processin has been successfully used as a deep prior for seismic sh t athers interpolation (Kong et al., 2020; Liu al., 2019). Here, we use a modified arc itecture based the rationale that se smic data exhibit self-similari at different scales and that the output of the network (i.e. the reconstructed time slice) should symmetric. The main points of the modified architecture are the following: • in order to captur the f atu es of seismic data at different cales we use the multi-resolution bl c A sequence of 3×3 onv lu al layers extracts fe tures at different scales, and their outputs a concatenated together. Moreover, a residual connection with 1×1 convolutio , which proved t effective for interpolation problems, is added; • th number o fil ers in the sequence is g adually ncre sed, in rder to efficiently manage memory • the rect skip-connection, disti ctiv feature of stan ard U-Net, is replaced by the path residu block which includes the 3×3 a d 1×1 convoluti ns. Its n -li ear t ansformations on encod features balance the semantic gap introduced by deeper d coder stag ; • downsampling is performed by 3×3 convolutions with stride 2×2, upsampling is performed bilinear interp la io instead of deconvolutions, improved the results; , which is an ill-posed inverse problem. We solve this problem through the deep image prior paradigm (Ulyanov e al., 2018), where the CNN archi ecture itself can capture enough low-level features (i.e., our prior) from a single corrupted seismic dat volum , mainly exploiting the self-similarities in t da a. Through this paradigm, th CNN is model d as a param tric func on M ( x S , x R ) = 1 if x S , x R 0 otherw Therefore, the dataset X 0 acquired with the actual acquisiti application of the mask M to the ideal survey X for each time s case n which t e act al n mber f sources is smaller than th dataset. To recover X starting from X 0 , we can invert for X the equa inv rse problem. We solve this problem through t deep imag where the CNN architecture itself can capture enough low-lev corrupted seismic data volume, mainly exploiting the self-simila he CN is mod led as a param tric unc i ^ X = F θ ( Z ) mappin The weights θ are randomly initializ d and updated by minim zi this ase the distance between the generated d ta ^ X and the origi θ = argmin θ ‖ M ⨀ F θ ( Z ) − Wh le the loss is computed only on he actual acqu sition geomet whole ideal survey: the network learns how to transform a fixed from the decim ted data itself; the missing traces are r stored at t M ULTI - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-net architecture (Ronneberger et al., 2015), which has bee has been suc essfully u ed as a deep prior for seismic sho gathe al., 2019). Here, we use a modified architecture based the rationa at different sc les and ha the ou put of the network (i.e. t symmetric. The main points of the modified architecture are the f • in order to captur the features f seismic data at different A sequence of 3×3 convolutional layers extracts features concatenated together. Moreover, a residual connection w effective for interpolation problems, is added; • t e number of filters in the sequen e is gradually increased • the direct skip-connection, distinctive feature of standard block which includes th 3×3 an 1×1 convolutions. It features balance the semantic gap introduced by deeper de • downsampling is performed by 3×3 convolutions with bil ear int rpolations instead of deconvolutions, improved mapping a noise realization M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise ataset X 0 acquired with the actual acquisition geometry G can be modeled as the e mask M to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We consider the e actual number of sources is smaller than the number of sources in the ideal dense arting from X 0 , we can invert for X the equation X 0 = M ⨀ X , which is an ill-posed . We solve this robl m through the deep image p ior paradigm (Ulyanov t al., 2018), architecture itself can capture enough low-level features (i.e., our prior) from a single c data volume, mainly exploiting the self-si ilarities in the data. Through this paradigm, eled as a parametric function ^ X = F θ ( Z ) mapping a noise realization Z to the data space. e randomly initialized and updated by minimizing a loss function on the known traces, in nce between the generated data ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 computed only on the ctual acquisition geometry, the output of the CNN ranges over the ey: the network learns how to transform a fixed noise volume Z into the decimated data ted data itself; the missing traces are restored at the same time. TION U-N ET WITH SYMMETRIC OUTPUT ecture (Ronneberger et al., 2015), which has been designed for medical image processing, fully used as a deep prior for i mic shot gathers inte polation (Kong et al., 2020; Liu et we use a modified architecture based th rationale that seismic data exhibit self-similarity es and that the output of the network (i. . the reconstructed time slice) should be main points of the modifi d archit cture are the following: capture the features of seismic data at differe t scales we use the multi-res lution block. e of 3×3 convolutional layers extracts features at different scales, and their outputs are ed together. Moreover, a residual connection with 1×1 convolution, which proved to be or interpolation problems, is added; r of filters in the sequence is gradually increased, in order to efficiently manage memory; skip-connection, distinctive feature of standard U-Net, is replaced by the path residual ch includes the 3×3 and 1×1 convolutions. Its non-linear transformations on encoder lance the semantic gap introduced by deeper decoder stages; ling is performed by 3×3 convolutions with stride 2×2, upsampling is performed by terpolations instead of deconvolutions, improved the results; to the data space. The weights M ( x S , x R ) = Therefore, the dataset X 0 acqu red with the a application of the mask M o the ideal survey X case in which the actual number of sources is s dataset. To recover X starting from X 0 , we can invert f inverse problem. We solve this problem through where the CNN arch tecture tself c n capture e cor upted seismic data volume, mainly exploiti g t e CNN is modeled as parametric function ^ X The we ghts θ are randomly initialized and update this case the distance between the generated data θ = argmin θ While the loss is computed only on the actual acq whole id al survey: the network learns how to tra from the decimated data itself; the missing traces M ULTI - RESOLUTION U-N ET WITH SYMMETRI T e U-net architectu e (Ronn berger et al., 2015), has been suc essfully used as a deep prior for seis l., 2019). Here, we us a modified architecture ba at different scales and that the output of the sy metric. The mai poi ts of the modified archit • in order to capture the features of sei mic d A sequence of 3×3 convolutional layers ex concatenated together. Moreover, a residua effectiv for int rpolation problems, i adde • the number of filters in the sequence is grad • the direct skip-connection, distinctive feat block which includes th 3×3 and 1×1 c features bal nce the semantic g p intro uce • downsampling is performed by 3×3 conv bilinear interpolations instead of deconvolut are randomly initialized and u dated by minimizing loss function on th know traces, in this case thedistance between the generated data M ( x S , x R ) = 1 if x S , x R G 0 otherwise X 0 acquired with the actual acquisition geometry G can be modeled as the M to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We consider the al number of sources is smaller than the number of sources in the ideal dense from X 0 , we can invert for X the equation X 0 = M ⨀ X , which is a ill-posed lve this problem through the deep image prior paradigm (Ulyanov et al., 2018), cture itself can capture nough low-level features (i.e., our prior) from a single volume, mainly exploiting th self-similaritie in the data. Through this paradigm, a parametric function ^ X = F θ ( Z ) mapping a noise realiz tion Z to the data spa e. mly initialized and updated by minimizing a loss function on the known traces, in tw en the e erat t ^ X and the orig nal data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 ted only on the actual acquisition geom try, the output of the CNN ranges over th network learns how to transform a fixed no se volume Z into the decimated data itself; the missing traces are rest red a the sam tim . -N ET WITH SYMMETRIC OUTPUT Ronneberger et al., 2015), which has been designed for medical image processing, sed as a deep prior for seismic shot gathers interpolation (Kong et al., 2020; Liu et a modified architecture based the rationale that seismic data exhibit self-similarity that the output of the n twork (i.e. the reconstructe time sl ce) should be ints of the modified archit cture are the following: e the features of s ismic data at differ nt scales we u e the multi-resolution block. 3 convolutional layers extracts f atures at diffe ent scales, and their utputs are ther. Mor over, a residual connection with 1×1 volution, which proved t be polation problems, is add d; ers in the sequence is gradually incr ased, in order to efficiently ma ag m mory; nne tion, di tinctive feature of standard U-Net, is repla ed by the path r sidual udes the 3×3 and 1×1 convolutions. Its non-linear transf rmations on n der he semantic gap intr duced by deeper decoder stag s; perform d by 3×3 onvolutio s with str d 2×2, upsampling is performed by ions instead of deconvolutions, improved the results; d th original dat Any acquisition geometry G ( x S , x R ) is defined by a N ×N binary matrix M ( x S follows: M ( x S , x R ) = { 1 if ( x S , x R ) ∈ G 0 otherwise Therefore, the dataset X 0 acq ired with the ac ual acquisit geometry G can application of he m sk M to the ideal survey X for e ch tim slice, i.e., X 0 = M ⨀ case in which the ac ual number of sources is sm ller than the number of source dataset. To recover X starting from X 0 , we can inv rt for X the equation X 0 = M ⨀ X , invers problem. We solve this problem through the deep imag prior p radigm (U where the CNN architecture itself can capture enough low-level features (i.e., our corrupted seismic data volume, mainly exploitin he self-similarities in the data. Th the CNN is modeled as a para tric function ^ X = F θ ( Z ) mapping a nois realization The weights θ are randomly i itialized and updated by minimizing a loss function on this ase he istanc be ween he generat d dat ^ X and the original data X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 While the loss s computed only on th ctual a quisition geometry, the ou put of the whole id al survey: the network learns h w o transform a fixed noise v lume Z int from the decimated data itself; the mis ng traces re restore at the same time. M ULTI - RESOLUTION U-N ET WITH SYMMETRIC OUTPUT The U-net architecture (Ronneberger et al., 2015), which has been designed for medi has be successfully used as a deep prior for seismic shot gathers interpolatio (Ko al., 2019). Here, w use a modified architecture bas d the rationale that seismic data at different scales and that the utput of the network (i.e. the reconstruct d ti symmetric. The m in points of the mod fied architec ur are the following: • in order t capture the features of seismic data at different scales we use the m A sequence of 3×3 convolutional layers extracts features at different scales, concatenated together. Moreover, a residual connection with 1×1 convolution effective for inte pol tion problems, is ad ed; • the number of filters in the sequence is gradually increased, in order to efficient : S R 0 otherwise 0 acquired with the actual acquisition geometry G can be modeled as the to the ideal survey X for each time slice, i.e., X 0 = M ⨀ X . We consid the umber of sources is smalle than the number of s urces in the ideal dense X 0 , we can invert for X the equation X 0 = M ⨀ X , which is an ill-pos d this problem through the deep image pr r paradigm (Ulyanov et al., 2018), re itself can capture enoug low-level features (i.e., ur prior) from a single me, main y expl iting the self-similarities in th da a. Through this paradigm, arametric funct on ^ X = F θ ( Z ) mapping no e realization Z t the data space. y initi lized and updated by minimizing a loss function on the known traces, in en the generated data ^ X an the original dat X 0 : θ = argmin θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 only on the actual acquisition geometry, the output of the CNN ranges over the work ear s ow to transfor a fixed noise volume Z into the decimat d data elf; t e mi sing traces are restored at the same time. ET WITH SYMMETRIC OUTPUT nneberger et al., 2015), wh ch has been d signed for medic l im ge processing, as a d ep prior for seismic s ot gathers i terpolation (Kong et al., 2020; Liu et odified architecture bas d the rati nale that seismic data exhibit self-similarity at the output of the network (i.e. the r constructed time slice) hould b s of the modified architecture are the following: e features of seism c ata at different sc les we us the multi-resolution block. onvolutional layers extracts fe tures at different scal s, and their outputs are r. More v r, a r sidual connecti n with 1×1 onvolution, which prov d o be a ion probl ms, is adde ; in the sequence is gradually incr ased, in order to efficiently manage memory; ction, d stinctive feature of st n ard U-Net, is repl c d by the path residual s the 3×3 and 1×1 c nvolutions. Its non-linear transformat ons o encoder emantic gap introduce by deeper decoder stages; rformed by 3×3 convolutions with stride 2×2, upsampling is performed by s instead of deco volutions, improved the results; Whi e the loss is computed only n the actual acquisition geometry, the output of the CNN ranges over the who e id al survey: the network learns how t tran f rm a fixed ois volume fined by N ×N binary matrix M ( x S , x R ) (i.e., a mask) as , x R ) = { 1 if ( x S , x R ) ∈ G 0 oth rwise the actual acquisition geo et y G can be modeled as t e ey X for each time slice, i.e., X 0 = M ⨀ X . We consid r the s is smaller than the number of sources in the ideal dense vert for X th equation X 0 = M ⨀ X , wh h is an ill-posed rough he deep image prior pa adigm (Ulyanov t l., 2018), ure enough low-level features (i.e., our prior) from a single oiting the self-similarities in the data. Through this paradigm, n ^ X = F θ ( Z ) mapping a noise realization Z to the dat space. updated by minimizing a loss function on the known traces, in data ^ X and the original data X 0 : gmi θ ‖ M ⨀ F θ ( Z ) − X 0 ‖ 1 al acquisition geo e ry, the output of the CNN ranges over the to transform a fix d noise volume Z into th decimated data races re stored at the same time. ETRIC OUTPUT 2015), wh ch has been desig d f r medical im ge proc ssing, or seismic shot gath r interpol tion (Kong et al., 2020; Liu t ture based the rationale that sei mic data exhibit self-similar ty the network (i.e. the reconstructed time slice) should be arch tecture ar the followin : mic data at differen scales we us th m l i-resolution block. ers xtracts features at different scales, and their outputs are esidual connection with 1×1 convolution, w ich prove to be added; gradually increased, in order to eff ciently manage memory; f ature of standard U-Net, is r placed by the path resi ual ×1 conv lutions. Its n -linear transformations on encoder oduced by deeper decoder stages; convolutions with stride 2×2, upsampling is performed by nvolutions, improved the results; into the decimated data from the decimated dat itself; the missing traces are restored at the same time. Multi-resolution U-Net with symmetri o put The U-net architecture (Ronneberger et al., 2015), which has been designed for medical ima e proce sing, has b en successfully us d as a de p prior for eismic shot g ther interpolati n (Kong et al., 2020; Liu et al., 2019). Here, we use a modified architecture based the rationale that seismic data exhibi self-similarity at differ nt scales and t at the output of the netw k (i.e. the reconstru ted time lic ) sho ld be ymmet ic. The main points of the m dified architecture are the following: • in order to capture the fea ures of seismi at at d fferent scales we use t e multi- resolution block. A sequence f 3×3 convolutional layers ex acts features at different scales, and their outputs are con at nated together. M reover, a residual connection with 1×1 convolution, which proved to be effective for interpolati n problems, is added; • the number of filters in the sequence is gradually increase , in ord r to fficiently man ge memory; • the direct skip-connection, distinctive feature of standard U-Net, is replaced by the path residual blockwhich includes the 3×3 and 1×1 convolutions. Its non-linear transformations on encoder features bala ce the semanti gap in roduced by de per decoder stages; • ownsampling is performed by 3×3 c nvolutions with strid 2×2, upsampling is per orm d by bili ear interpolations in tead of deco volutions, improved the results; • The n twork kernels are extend d to 3D convolutions, whic captur corr lations between adjacent tim slic s; • the nal symmetry block is designed to generate a symm tric output. Being • The network kernels are extended to 3D convolutions, which capt time slices; • the final symmetry bl ck is designed to generate a symmetric o symmetrizing bl ck, the output i obtained as ( Y + Y T )/ 2 . This bl prior into the network architecture forcing reciprocity between sour Notice that Bat h Normalization an Leaky Rectified Linear Unit convolu ional filters but the last one, esponsible for the generation of full The input tensor Z is a Gaussian noise realization whose size is the sam used Adam optimizer with learning rate of 0.001 in each experimen the i put of the symmetrizing block, the output is obtained as • The network kernels are exte ded to 3D co volutions, which captu e cor elations betw time slices; • the final symmetry block is designed to generate a symmetric output. Being Y the symmetrizing block, the output is obtained as ( Y + Y T )/ 2 . This block introduces a ph prior into the network architecture forcing reciprocity between source-receiver pairs. Notice that Batch Normalization and Leaky Rectifi d Li ear Units are implemented conv luti nal filters but the last one, responsible for the generation of fully sampled time slice The input tensor Z is a Gauss an noise realization whose size is the same as the desired out . This block introduces a physics-derived prior i to h etwork architecture f rcing recip ocity between source- receiver pai s. Notice that Batch Normalization and Leaky Rectified Linear Units are implemented for all the convolutional filters but the last one, responsible for the generation of fully sampled time slices.