Extrapolated full waveform inversion with deep learning

2 Introduction

• Because seismic inversion data are typically limited to frequencies above 3Hz, FWI may lack the low-frequency information needed to avoid local minima from poor initial models.The paper notes that existing low-frequency recovery approaches remain far from the true low-frequency data.Source: §2 · p. 2
• Earlier signal-processing approaches reconstruct isolated spikes, while more recent methods recover low frequencies that remain far from the true data.The cited approaches include envelope methods and reflectivity-series inversion followed by convolution with a broadband source wavelet.Source: §2 · p. 2
• The proposed DNN learns to recover missing low frequencies directly from raw bandlimited records, unlike methods that explicitly parameterize event phases and amplitudes.The CNN operates in the time domain and can use one trace or multiple traces to predict a low-frequency band.Source: §2 · pp. 2–3
• Training uses synthetic wide-band shot gathers from different models to regress low-frequency data from high-frequency data; testing uses unprocessed bandlimited records from a new scenario.In synthetic tests, the wave solver can provide the low frequencies for validation.Source: §2 · pp. 2–3
• The authors identify insufficient training data, lack of physical interpretation, and the absence of a theory explaining deep-network generalizability as limitations.Despite these concerns, their numerical examples indicate generalization to unknown subsurface structures across a range of experiments.Source: §2 · p. 3
• On the BP 2004 model, extrapolated low frequencies enabled conventional FWI from an uninformative initial model without cycle-skipping at 0.6Hz.The network also showed acceptable robustness to noise, a poorly known source wavelet, and different finite-difference schemes.Source: §2 · p. 3

3 Deep Learning

• Each network layer composes affine transformation, batch normalization, and nonlinear activation; experiments found PReLU works better than ReLU for bandwidth extension.The network’s depth is the length of this function chain.Source: §3 · p. 4
• The training objective minimizes squared error between true low frequencies and network estimates using mini-batch stochastic gradient descent and backpropagation.The cost function is minimized over the network parameters.Source: §3 · p. 4
• The CNN treats each time-domain seismic amplitude as an input pixel, with convolution exploiting local correlations and global features through local connections and weight sharing.The network is used for regression of continuous-valued outputs rather than classification labels.Source: §3 · pp. 4–5
• Pooling is omitted because input and output signals have equal length, making feature-map downsampling unhelpful for bandwidth extension.Batch normalization can speed training and reduce sensitivity to network initialization.Source: §3 · p. 5
• Large convolution filters increase the receptive field quickly with depth, giving the network freedom to reconstruct long-wavelength information.The receptive field is the input region that affects a neuron’s response.Source: §3 · p. 5
• The architecture stacks five convolution–PReLU–batch-normalization blocks before a fully connected layer outputs low-frequency time-domain amplitudes.The convolutions use stride one and zero-padding to preserve signal length; dropout follows the first convolution.Source: §3 · p. 5, Figure 1 · p. 6
• Training samples pair high-frequency bandlimited seismograms with their low-frequency counterparts, and models should span varied subsurface structures to support generalization.The network is trained on many pairs before testing on new inputs.Source: §3 · p. 6

4.1 Low frequency extrapolation

• On Marmousi, the CNN reconstructed missing 0.1–5Hz recordings from 5–20Hz inputs, matching true low-frequency traces in amplitude and phase and recovering reflections after direct waves were muted.Training data came from nine structurally varied Marmousi submodels; training and test losses both declined over 20 epochs.Source: §4.1 · pp. 7, 10, Figure 8 · p. 13, Figure 9 · p. 14

4.2 Uncertainty analysis

• With 20% test and 30% training noise, RMSE was 2.4156×10−4; fourth-to-sixth-order FD transfer yielded 2.2248 × 10−4, while Ricker-to-Ormsby transfer raised RMSE to 1.1717 × 10−3 and weakened amplitude recovery.Despite the larger error with the Ormsby wavelet, phase extrapolation remained comparatively robust; the authors note that broader training-data diversity could improve results.Source: §4.2 · pp. 14–15

4.3 Extrapolated FWI: Marmousi model

• On Marmousi, extrapolated-low-frequency initialization produced a bandlimited-FWI result nearly matching the true-low-frequency initialization, while an uninformed linear start suffered pronounced cycle skipping.Both low-wavenumber starting models captured the model’s broad structure, but recovery below 2km remained difficult; inversion from the linear initial model fell into a local minimum.Source: §4.3 · p. 20, Figure 15 · p. 21

4.4 Extrapolated FWI: BP model

• With 0.6–2.4Hz data, extrapolated-low-frequency initialization overcame cycle skipping on BP and outperformed bandlimited-only FWI; it still reduced model error when available data began at 1.2Hz.The network trained on Marmousi submodels generalized to BP’s salt structure. Its performance decreased as the lowest available frequency rose, and excessive iterations with extrapolated data could worsen the inversion; accuracy at 0.3Hz varied little between input bands [0.6, 20]Hz and [1.2, 20]Hz.Source: §4.4 · pp. 22, 24, 25, Figure 21 · p. 27
• The BP-model comparison contrasts FWI initialized from the original model with FWI initialized from an inverted low-wavenumber model using extrapolated 0.3Hz, 0.6Hz, and 0.9Hz data.Those below-1.2Hz data were recovered from 1.2–20Hz bandlimited data.Source: Figure 23 · p. 29
• EFWI-CNN improves inverted-model quality over FWI using only bandlimited data across the tested lowest input frequencies.The comparison also includes FWI using fullband data; the passage does not state that EFWI-CNN outperforms it.Source: Figure 24 · p. 30
• EFWI-CNN performance declines as the lowest frequency of the bandlimited data increases, with inversion’s highest frequency fixed at 2.4Hz.Source: Figure 24 · p. 30

Training limitations

• An incomplete training set or unsuitable architecture can cause large generalization error, and the small-data, high-capacity example is prone to overfitting.Deep-learning training is demanding because its time depends on dataset size and model capacity; downsampling inputs and outputs followed by bandlimited interpolation is proposed to reduce network weights.Source: §5 · pp. 30–31

Data imbalance

• Reflected-wave extrapolation is less accurate than primary-wave extrapolation in this example because strong direct-wave energy biases fitting toward the direct wave.Source: §5 · p. 31

Multi-trace extrapolation

• Multi-trace input improves coherence across traces and helps reduce random noise, but it does not help correlated noise.Figure 25 compares one-, five-, and seven-trace inputs on the full-size Marmousi model.Source: §5 · p. 31, Figure 25 · p. 32

Frequency range and model scope

• Generating BP 2004 data in the broader [0.1, 1.4]Hz band remains an open architecture, hyperparameter, and training-schedule problem after results in [0.1, 0.5]Hz.Other community models and more realistic physics, including elastic waves, remain to be explored.Source: §5 · p. 32

Field-data challenges

• The effect of training-test physics differences remains unstudied, limiting what is established for field data with correlated and uncorrelated noise, shot-varying unknown wavelets, viscoelasticity, and anisotropy.Source: §5 · pp. 32–33

• The CNN extrapolates missing low frequencies from bandlimited data without input preprocessing or output post-processing.Source: §6 · p. 33
• Extrapolated low frequencies can provide a reliable low-wavenumber velocity model for initializing FWI on bandlimited data without cycle skipping.Source: §6 · p. 33
• A large receptive field is necessary for low-frequency extrapolation, while architecture and hyperparameters can be adjusted to the bandlimited data.Source: §6 · p. 33