Source-linked AI summary
Semi-supervised Sequence Modeling for Elastic Impedance Inversion
Motaz Alfarraj, Ghassan AlRegib
TL;DR
Learning-based seismic inversion often has limited training data and rarely enforces geophysical constraints. This paper combines recurrent and convolutional sequence modeling with well-log supervision and seismic forward modeling for multi-angle EI inversion. On a synthetic model, estimated EI has an average PCC of approximately 98%.
Problem
Learning-based seismic inversion faces scarce well-log labels and often does not enforce a physical model, while seismic inversion is highly ill-posed.
Source: INTRODUCTION · pp. 2–3Method
A semi-supervised recurrent-and-convolutional sequence model inverts multi-angle seismic data for EI using well logs and seismic forward modeling.
Source: INTRODUCTION · p. 3, PROBLEM FORMULATION · p. 4Results
Average PCC ≈98% and average r2 ≈94% for estimated EI on the synthetic section.
Source: Results and Discussion · p. 10Takeaways & Limitations
The estimated EI captures the overall trend of true EI in the reported synthetic case using 10 well logs for training.
Source: INTRODUCTION · p. 3, Results and Discussion · p. 10The reported Marmousi 2 model has 2720 traces, and computation time scales linearly with the number of dataset traces.
Source: Implementation · p. 13
Abstract
from arXiv · showhide
Recent applications of machine learning algorithms in the seismic domain have shown great potential in different areas such as seismic inversion and interpretation. However, such algorithms rarely enforce geophysical constraints - the lack of which might lead to undesirable results. To overcome this issue, we have developed a semi-supervised sequence modeling framework based on recurrent neural networks for elastic impedance inversion from multi-angle seismic data. Specifically, seismic traces and elastic impedance (EI) traces are modeled as a time series. Then, a neural-network-based inversion model comprising convolutional and recurrent neural layers is used to invert seismic data for EI. The proposed workflow uses well-log data to guide the inversion. In addition, it uses seismic forward modeling to regularize the training and to serve as a geophysical constraint for the inversion. The proposed workflow achieves an average correlation of 98% between the estimated and target EI using 10 well logs for training on a synthetic data set.
INTRODUCTION
The paper addresses sparse well-log supervision and limited use of physical constraints in learning-based seismic inversion. It proposes semi-supervised sequence modeling for multi-angle elastic impedance inversion, combining recurrent and convolutional networks with a seismic forward model.
INTRODUCTION
- The proposed workflow inverts multi-angle seismic data for EI using recurrent and convolutional networks to model traces as sequences.It learns from well-log data and corresponding seismic data, with a forward model guiding and regularizing learning.Source: INTRODUCTION · p. 3, PROBLEM FORMULATION · p. 4
PROBLEM FORMULATION
- The workflow combines well-log supervision with seismic data misfit, allowing learning to use all survey traces while only some have EI labels.The seismic loss is computed over all traces, whereas property loss uses traces with well-log measurements.Source: PROBLEM FORMULATION · pp. 3–4
RECURRENT NEURAL NETWORKS
- GRUs are used to capture long-term dependencies through update and reset gates that control information flow across time steps.The paper notes that ordinary RNNs can fail to learn long-term dependencies when gradients vanish across long sequences.Source: RECURRENT NEURAL NETWORKS · p. 5
METHODOLOGY
- A seismic forward model provides an additional form of supervision, incorporates geophysics into learning, and permits training on traces without EI labels.The method is motivated by limited well-log data, which constrains the number of training samples and network layers in supervised learning.Source: METHODOLOGY · pp. 5–6
Proposed workflow
The workflow combines an inverse model with a fixed seismic forward model, using well-log property loss and seismic data misfit to train the inverse mapping. The seismic loss extends training to traces without corresponding property logs.
Proposed workflow
- The inverse model estimates EI from multi-angle seismic traces, and a fixed forward model synthesizes seismograms for comparison with the input traces.The combined seismic and property losses adjust the inverse-model parameters by gradient descent.Source: Proposed workflow · p. 6
- The seismic data misfit trains on all survey traces, while property loss compares estimated and true EI only where well-log data are available.The workflow can therefore learn from seismic traces without corresponding property traces, alongside the few labeled well-log traces.Source: Proposed workflow · p. 6
- The two losses use mean squared error with α = β = 1, and the seismic loss covers many more traces than the property loss.The authors note that the direct well-log property data and broader seismic coverage contribute in tandem to learning the inverse model.Source: Proposed workflow · p. 6
- The forward model must be fast and differentiable; the authors use a convolutional model for its simplicity and efficiency.Other forward models are possible if they satisfy both conditions.Source: Proposed workflow · p. 6
Inverse Model
The inverse model combines recurrent sequence modeling, dilated convolutions, upscaling, and regression to estimate EI from seismic traces. Its modules target different frequency content and account for the resolution difference between seismic and well-log data.
Inverse Model
- The inverse model has four submodules: sequence modeling, local pattern analysis, upscaling, and regression, each with a distinct task.The modules' features are combined, upscaled, and mapped to EI.Source: Inverse Model · p. 6
Sequence Modeling
- Three bidirectional GRUs model temporal dynamics from past and future trace samples, producing features treated as the low-frequency EI trend.The first GRU takes the incident-angle channels as inputs; the three GRUs form a deep sequence model.Source: Sequence Modeling · pp. 6–7
Local pattern analysis
- Parallel dilated convolutions extract multiscale local attributes for high-frequency EI trends, complementing recurrent features that capture low-frequency content.The local and sequence-modeling outputs are added to obtain a full-band representation.Source: Local pattern analysis · p. 7
- Two deconvolutional blocks compensate for lower seismic sampling resolution, then a GRU and linear layer map features to the EI target domain.The regression GRU augments upscaled outputs using global temporal features.Source: Inverse Model · p. 6, upscaling · p. 8, Regression · p. 8
Forward Model
The forward model synthesizes seismograms from EI using reflection coefficients and wavelet convolution. For the Marmousi 2 case study, the authors generate synthetic EI and seismic data and compare inverted EI with the known model.
Forward Model
- The forward model computes reflection coefficients from normalized EI using Connolly’s formulation, then convolves them with a wavelet to synthesize seismograms.Its input is EI and its output is the corresponding seismogram.Source: Forward Model · p. 8
- The Aki-Richards approximation used for elastic impedance is valid only below 35° incident angle, so the case study considers only valid angles.EI depends on P-wave velocity, S-wave velocity, density, and incident angle; the normalized formulation uses training-sample averages.Source: Forward Model · p. 8
CASE STUDY ON MARMOUSI 2 MODEL
- Marmousi 2 spans 17 km by 3.5 km at 1.25 m vertical resolution; its elastic model is converted to time to generate EI and seismic data.The authors invert the full model and compare estimated EI with true EI quantitatively.Source: CASE STUDY ON MARMOUSI 2 MODEL · p. 8
Dataset Generation
The synthetic dataset uses four incident angles and 2,720 seismic traces generated from EI. Ten evenly spaced traces provide paired EI and seismic data for training; the remaining traces provide seismic data only.
Dataset Generation
- Four incident angles—0°, 10°, 20°, and 30°—are used to generate EI from the Marmousi 2 elastic model.The seismic inputs have four channels, one per angle.Source: Dataset Generation · p. 8
- 2,720 seismic traces are generated from EI with a 5-10-60-80 Hz Ormsby wavelet, downsampled sixfold, and given 15 dB white Gaussian noise.The noise is added to assess workflow robustness.Source: Dataset Generation · p. 9
- Ten evenly spaced traces provide paired EI and seismic training data; all other survey traces have seismic data only.Figure 6 shows the training EI traces overlaid on the zero-offset seismic section.Source: Dataset Generation · p. 9, Figure 6 · p. 9
Training The Inverse Model
The inverse model is trained on randomly selected seismic traces and traces paired with well-log EI, with a forward model providing a seismic-consistency loss. The workflow uses cross-validation to select some model hyperparameters while setting others from data requirements.
Training The Inverse Model
- Training repeats inverse prediction, forward synthesis, loss calculation, and parameter updates until convergence.The total loss sums seismic MSE between synthesized and input seismic traces with property MSE on training traces that have true EI.Source: Training The Inverse Model · p. 9
- The seismic loss uses randomly selected traces and well-log-paired traces, while property loss is computed only on traces with known EI.The forward model synthesizes seismograms from estimated EI; the property loss compares predicted and true EI traces.Source: Training The Inverse Model · p. 9
- The input-channel count is set to the number of incident angles, and the two stride factors are chosen so their product matches the seismic-to-EI resolution mismatch.For this case study, c1 = 4, s1 = 3, and s2 = 2; other hyperparameters were selected by validation performance using cross-validation.Source: Training The Inverse Model · p. 9
Results and Discussion
On Marmousi 2, the estimated EI captures the true EI trend, with average PCC ≈98% and average r2 ≈94%. The analysis also reports localized errors and lateral jitter, alongside trace- and image-based evaluations.
Results and Discussion
- Estimated EI is accurate across most of the section but shows visible lateral jitter, attributed to 1D sequence modeling without explicit spatial constraints and to seismic noise.The displayed sections are direct workflow outputs without post-processing to reduce artifacts.Source: Results and Discussion · p. 9
- At x = 3200 m, an EI anomaly causes an incorrect estimate, but the GRU reset gate limits error propagation and estimates recover at shallower and deeper samples.At x = 8500 m, the estimate follows the true trace trends, including thin layers.Source: Results and Discussion · p. 9
- Most estimated EI samples lie within one standard deviation of true EI and show a linear correlation across incident angles.Figure 10 shades the region within ±σEI of true EI.Source: Results and Discussion · p. 9, Figure 10 · p. 12
- PCC measures linear correlation without regard to individual values, whereas r2 accounts for mean squared error and M-SSIM assesses section similarity using local image statistics.M-SSIM averages SSIM scores over local windows; its image-based calculation reflects lateral and vertical variation.Source: Results and Discussion · pp. 9–10
- Average PCC ≈98% and average r2 ≈94% indicate that estimated EI captures the overall trend of true EI on Marmousi 2.PCC and r2 are averaged across traces for each incident angle; Table 1 summarizes the quantitative evaluation.Source: Results and Discussion · p. 10, Table 1 · p. 12
Implementation
The workflow was implemented with Adam and converged in about 300 iterations; on Marmousi 2, it estimated EI with an average correlation of 98% and is described as extendable to broader inversion and reservoir-property estimation.
Implementation
- Training used Adam and converged in about 300 iterations, though loss continued to decrease slightly before training stopped at 500 iterations to avoid overfitting.The reported GPU-accelerated runtime for 500 iterations was 2 minutes.Source: Implementation · pp. 12–13
- Efficiency relies on 1-dimensional modeling and a relatively small Marmousi 2 model of 2720 traces, while computation time scales linearly with dataset trace count.Source: Implementation · p. 13
- The code used to reproduce the reported results is publicly available on GitHub.Source: Implementation · p. 13
CONCLUSIONS
- 98% average correlation was achieved for EI estimates on Marmousi 2 using a small number of training EI traces.The workflow was validated on the Marmousi 2 model.Source: CONCLUSIONS · p. 13
- The workflow can be extended to full elastic inversion and property estimation for reservoir characterization.Source: CONCLUSIONS · p. 13