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In subsurface imaging, learning the mapping from velocity maps to seismic waveforms (forward problem) and waveforms to velocity (inverse problem) is important for several applications. While traditional techniques for solving forward and inverse problems are computationally prohibitive, there is a growing interest in leveraging recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data.
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Despite the variety of architectures explored in previous works, several open questions remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems.
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We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward-Inverse (GFI) framework, building on the assumption of manifolds and latent space translations.
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We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI.
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We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain translation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively.
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We show that our proposed models achieve state-of-the-art performance for forward and inverse problems on a wide range of synthetic datasets and also investigate their zero-shot effectiveness on two real-world-like datasets.
We propose Generalized Forward-Inverse (GFI) framework based on two assumptions. First, according to the manifold assumption, we assume that the velocity maps v ∈ v𝒱 and seismic
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waveforms p ∈ p𝒱 can be projected to their corresponding latent space representations, ṽ and p̃, respectively, which can be mapped back to their reconstructions in the original space, v̂ and p̂.
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Note that the sizes of the latent spaces can be smaller or larger than the original spaces. Further, the size of ṽ may not match with the size of p̃. Second, according to the latent space
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translation assumption, we assume that the problem of learning forward and inverse mappings in the original spaces of velocity and waveforms can be reformulated as learning translations in their
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latent spaces.
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</p>
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<ol>
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<li>
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<b>Latent U-Net Architecture: </b> We propose a novel architecture to solve forward and inverse problems using two latent space translation models implemented using U-Nets, termed Latent U-Net. Latent
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U-Net uses ConvNet backbones for both encoder-decoder pairs: <code>E</code><sub>v</sub>, <code>D</code><sub>v</sub> and <code>E</code><sub>p</sub>, <code>D</code><sub>p</sub>, to project
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v and p to lower-dimensional representations. We also constrain the sizes of the latent spaces of
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ṽ and p̃to be identical, i.e., dim(ṽ) = dim(p̃), so that we can train two separate U-Net models to implement the latent space mappings L<sub>ṽ → p̃</sub> and L<sub>p̃ → ṽ</sub>.
<b>Invertible X-Net Architecture: </b> We propose another novel architecture within the GFI framework termed Invertible X-Net to answer
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the question: “can we learn a single latent space translation model that can simultaneously solve
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both forward and inverse problems?” We employ invertible U-Net in the latent spaces of velocity and waveforms, which can be constrained to be of the same size (just like Latent-UNets), i.e.,
In subsurface imaging, learning the mapping from velocity maps to seismic waveforms (forward problem) and waveforms to velocity (inverse problem) is important for several applications. While traditional techniques for solving forward and inverse problems are computationally prohibitive, there is a growing interest in leveraging recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data.
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+
Despite the variety of architectures explored in previous works, several open questions remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems.
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+
We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward-Inverse (GFI) framework, building on the assumption of manifolds and latent space translations.
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+
We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI.
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We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain translation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively.
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We show that our proposed models achieve state-of-the-art performance for forward and inverse problems on a wide range of synthetic datasets and also investigate their zero-shot effectiveness on two real-world-like datasets.
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</p>
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<h2class="banded">Method Overview</h2>
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<p>
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We propose Generalized Forward-Inverse (GFI) framework based on two assumptions. First, according to the manifold assumption, we assume that the velocity maps v ∈ v𝒱 and seismic
94
+
waveforms p ∈ p𝒱 can be projected to their corresponding latent space representations, ṽ and p̃, respectively, which can be mapped back to their reconstructions in the original space, v̂ and p̂.
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+
Note that the sizes of the latent spaces can be smaller or larger than the original spaces. Further, the size of ṽ may not match with the size of p̃. Second, according to the latent space
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translation assumption, we assume that the problem of learning forward and inverse mappings in the original spaces of velocity and waveforms can be reformulated as learning translations in their
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latent spaces.
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</p>
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<ol>
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+
<li>
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<b>Latent U-Net Architecture: </b> We propose a novel architecture to solve forward and inverse problems using two latent space translation models implemented using U-Nets, termed Latent U-Net. Latent
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+
U-Net uses ConvNet backbones for both encoder-decoder pairs: (<code>E</code><sub>v</sub>, <code>D</code><sub>v</sub>) and (<code>E</code><sub>p</sub>, <code>D</code><sub>p</sub>), to project
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+
v and p to lower-dimensional representations. We also constrain the sizes of the latent spaces of
104
+
ṽ and p̃ to be identical, i.e., dim(ṽ) = dim(p̃), so that we can train two separate U-Net models to implement the latent space mappings L<sub>ṽ → p̃</sub> and L<sub>p̃ → ṽ</sub>.
<b>Invertible X-Net Architecture: </b> We propose another novel architecture within the GFI framework termed Invertible X-Net to answer
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+
the question: “can we learn a single latent space translation model that can simultaneously solve
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+
both forward and inverse problems?” We employ invertible U-Net in the latent spaces of velocity and waveforms, which can be constrained to be of the same size (just like Latent-UNets), i.e.,
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