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PSDWII: Physical-Structure-Driven Waveform Inversion Imaging

Shengchang Chen

arXiv:2609.03380v1physics.geo-ph

TL;DR

Multi-parameter FWI faces high computational cost and coupling, while optimization-based imaging methods can lack clear physical interpretation. PSDWII instead inverts virtual sources and develops three physical-structure-driven methods. The paper presents a unified theoretical framework whose core steps had been validated in earlier numerical experiments.

  • Problem

    Multi-parameter FWI is computationally costly, and optimization-based waveform inversion can obscure the physical meaning of its algorithms.

    Source: §1 · p. 1

  • Method

    PSDWII uses adjoint propagation to approximate virtual sources and removes incident-wavefield and radiation-pattern contributions to recover targets without objective functions or gradient and Hessian calculations.

    Source: §8 · p. 15

  • Results

    The paper presents a unified framework and three methods; their core steps had been validated in previous numerical experiments.

    Source: §1 · p. 3, §8 · p. 15

  • Takeaways & Limitations

    The framework uses radiation-pattern differences to support physical decoupling of multiple parameters, with effectiveness influenced by model parameterization.

    Source: §8 · p. 15

    In practice, deconvolution stability is challenged by band-limited data, noise, and limited illumination, while accuracy also depends on background-model traveltimes and acquisition constraints.

    Source: §7.7 · p. 14

Abstract

from arXiv · show

This paper proposes a Physical-Structure-Driven Waveform Inversion Imaging framework (PSDWII), grounded in the recognition that surface-recorded seismic data are directly related to subsurface virtual sources rather than to the model parameters themselves. Based on this recognition, we decompose seismic wave propagation into a three-step physical process: source excitation and incident wave propagation; interaction of the incident wave with heterogeneities, which generates virtual sources with distinct radiation patterns and excites secondary waves; and propagation and reception of the secondary waves. The adjoint of the propagation operator (reverse-time extrapolation) is then used to formulate a linear inversion projection for the virtual sources. By distinguishing different virtual-source expressions and their underlying physical mechanisms, we establish a unified mathematical representation for three inversion tasks: Full Waveform Inversion(FWI), stratigraphic physical properties imaging, and stratigraphic structures imaging. Deconvolution and linear inversion are respectively adopted to achieve multi-parameter decoupling in FWI and in angle-domain common-image gather inversion for stratigraphic physical properties imaging. The PSDWII framework shifts the conventional ``mathematical optimization driven'' paradigm of waveform inversion to a ``physical-structure-driven'' one. It neither constructs objective functions nor computes their gradients or Hessian inverses. Within this framework, we develop three specific methods: physical-structure-driven FWI (PSDFWI), physical-structure-driven stratigraphic physical properties imaging with angle-domain common-image gather inversion (PSDSI), and physical-structure-driven stratigraphic structures imaging (PSDMig).

1 Introduction

The introduction identifies physical interpretability and coupling challenges in optimization-driven seismic imaging, then proposes a virtual-source-based framework and three methods tailored to different inversion targets. This paper develops their theoretical foundations, while noting that core steps were validated in prior numerical experiments.

1 Introduction

  • Gradient-based multi-parameter FWI adds mathematical coupling to physical coupling, prompting computationally expensive Hessian-based decoupling.Source: §1 · pp. 1–2
  • Existing stratigraphic property and structure imaging lack suitable reflection wave equations, while Zoeppritz-based imaging formulas rely on horizontally or one-dimensionally layered media.Source: §1 · p. 2
  • PSDWII treats propagation as incident-wave input, virtual-source response, and recorded secondary-wave output, using reverse-time propagation to approximate the virtual source.Source: §1 · p. 2
  • The framework interprets the adjoint wavefield as an approximate inversion of the subsurface virtual source, giving the adjoint-state result a physical meaning.Source: §1 · p. 3
  • PSDFWI uses scattering virtual sources; PSDSI uses body-reflection virtual sources for property imaging and angle-domain gather inversion; PSDMig images surface-reflection virtual sources.Source: §1 · p. 3
  • The paper elaborates the unified theoretical framework; its core method steps had been validated in previous numerical experiments.Source: §1 · p. 3

2 Physical Process and Physical Structure of Seismic Wave Propagation

Seismic propagation is described as incident waves interacting with heterogeneities to create virtual sources, followed by secondary-wave propagation to receivers. The virtual-source type depends on heterogeneity scale, and boundary reflectivity is represented by a directional derivative.

2 Physical Process and Physical Structure of Seismic Wave Propagation

  • Propagation comprises incident-wave travel, virtual-source generation at heterogeneities, and secondary-wave travel to surface receivers.Source: §2 · pp. 3–4
  • Heterogeneities no larger than the dominant wavelength generate scattering virtual sources; larger ones generate reflection virtual sources characterized by opening angle θ.Source: §2 · p. 4
  • Local reflectivity of a reflector boundary is defined as the directional derivative of model-parameter perturbation along incident-wave propagation.Source: §2 · p. 5
  • Scattered and reflected waves share incident- and secondary-wave propagation, differing in the virtual source that generates them.Source: §2 · p. 5

3 Physical-Structure-Driven Seismic Waveform Inversion Imaging Framework (PSDWII)

PSDWII first approximates subsurface virtual sources from observed wavefields by adjoint propagation, then removes incident-wave and radiation-pattern contributions to recover target parameters. Its deconvolution relies on time consistency, and reconstructed incident fields support amplitude fidelity.

3 Physical-Structure-Driven Seismic Waveform Inversion Imaging Framework (PSDWII)

  • The framework first inverts the observed wavefield for a subsurface virtual source, then removes its incident-wavefield and radiation-pattern contributions to recover the target.Source: §3 · p. 6
  • Reverse-time adjoint propagation produces the approximate virtual-source inversion, giving the adjoint wavefield its physical interpretation.Source: §3 · p. 6
  • Deconvolution requires the inverted virtual source to be time-consistent with the incident wavefield; this consistency is necessary to identify it as the source of secondary waves.Source: §3 · p. 6
  • Using a reconstructed incident wavefield in the denominator supports good fidelity and amplitude preservation in PSDWII deconvolution.Source: §3 · p. 7
  • PSDWII avoids objective functions, gradients, and Hessian inverses, and unifies FWI, stratigraphic property imaging, and stratigraphic structure imaging.Source: §3 · p. 7

4 Physical-Structure-Driven Full-Waveform Inversion Method (PSDFWI)

PSDFWI iteratively models an incident wavefield, forms the receiver residual, estimates a scattering virtual source, and deconvolves its incident-wave and radiation-pattern contributions to update the model. Parameterization affects cross-talk, and prior numerical experiments demonstrated feasibility for velocity FWI.

4 Physical-Structure-Driven Full-Waveform Inversion Method (PSDFWI)

  • PSDFWI computes the initial-model wavefield and residual, estimates the scattering virtual source, removes incident-wave and radiation-pattern contributions, and iteratively updates the model.Source: §4 · pp. 7–8
  • Bulk-modulus and density parameterization is more conducive to reducing cross-talk than velocity and density parameterization because its radiation patterns have poorer coherence.Source: §4 · p. 8
  • PSDFWI avoids objective-function gradients and Hessian inverses for multi-parameter decoupling, and is described as lower-cost and lower-complexity than current FWI methods.Source: §4 · p. 9
  • Previous Marmousi-model numerical experiments demonstrated feasibility of PSDFWI for velocity inversion.Source: §4 · p. 9

5 Physical-Structure-Driven Stratigraphic Physical Properties Imaging and Angle-Domain Common-Image Gather Inversion (PSDSI)

PSDSI uses the primary reflection wavefield and a smooth model with accurate seismic kinematics to image stratigraphic property variations. Angle-domain common-image-gather inversion uses redundant angle information to refine property estimates and decouple parameter effects.

5.1 Fast Imaging of Stratigraphic Physical Property Variations (Relative Impedance Perturbations)

  • Fast PSDSI imaging produces angle-independent relative-impedance variation images from the primary reflection wavefield, given a smooth model with accurate kinematics.The fast image reflects combined property variations and can contain coupling between multiple parameters.Source: §5 · p. 9, §5.1 · p. 10
  • PSDSI directly uses the wave-equation-derived reflection virtual-source expression, with data requirements, computational conditions, and efficiency comparable to RTM, without a horizontally layered-medium assumption.Source: §5.1 · p. 9

5.2 Angle-Domain Common-Image Gather Inversion for Stratigraphic Physical Properties Imaging

  • Unlike Zoeppritz-based gather inversion, PSDSI derives its angle-varying property-perturbation expression from the wave equation and does not assume plane waves or horizontally layered media.Source: §5.2 · p. 10
  • Angle-domain inversion uses opening-angle-dependent radiation patterns and gather redundancy to linearly estimate velocity and density perturbations, which petrophysics can convert to lithologic perturbations.The workflow forms angle-domain common-image gathers from the inverted reflection virtual source and incident wavefield before solving across different angles.Source: §5.1 · p. 10, §5.2 · p. 11
  • Previous numerical experiments on acoustic random layered and Marmousi models validated PSDSI’s fast-imaging and angle-domain inversion workflows for stratigraphic physical-property parameters.Source: §5.2 · p. 11

6 Physical-Structure-Driven Stratigraphic Structures Imaging Method (PSDMig)

PSDMig images stratigraphic boundaries by estimating their local reflectivity from the primary reflection wavefield. It uses a smooth model with accurate kinematics and a reflection virtual-source expression derived from the wave equation.

6 Physical-Structure-Driven Stratigraphic Structures Imaging Method (PSDMig)

  • PSDMig images stratigraphic boundaries by imaging their local reflectivity from the primary reflection wavefield, given a smooth model with accurate seismic kinematics.Source: §6 · p. 11
  • The workflow constructs the incident wavefield, reverse-time extrapolates observed primary reflections to reconstruct the reflection wavefield, then images boundary local reflectivity.Source: §6 · p. 11
  • PSDMig directly uses a wave-equation-derived reflection virtual-source expression based on boundary local reflectivity, rather than Claerbout’s combination of heterogeneous-medium extrapolation and a simplified Zoeppritz imaging formula.Source: §6 · p. 11
  • Prior work completed comparisons with Claerbout’s cross-correlation imaging condition and imaging tests on point scatterers and reflecting surfaces.Source: §6 · p. 12

7 Discussion

The discussion distinguishes PSDWII from optimization by reversing the physical propagation chain to infer virtual sources and target parameters, and argues that this avoids gradient-induced mathematical coupling. It relates PSDMig to least-squares migration and FWI-Imaging, reviews method requirements and prior validations, and identifies theoretical assumptions and real-data stability as implementation boundaries.

7.1 Essential Difference Between Physical-Structure-Driven and Mathematical Optimization Driven Approaches

  • PSDWII reverses the propagation chain to extract virtual sources and remove the incident wavefield and radiation pattern, rather than optimize data mismatch through gradients.The framework follows the physical causal chain in reverse instead of constructing an objective function and using gradient descent.Source: §7.1 · p. 12

7.2 On the Problem of Multi-Parameter Coupling

  • Physical coupling jointly shapes wave propagation and carries subsurface information; gradient computation adds mathematical coupling, while PSDWII deconvolution can decouple it when radiation-pattern coherence is poor.The paper attributes added mathematical coupling to cross-correlation in the gradient method.Source: §7.2 · pp. 12–13
  • Bulk-modulus–density parameterization is more conducive to reducing cross-talk than velocity–density because its radiation patterns have poorer coherence.Source: §7.2 · p. 13

7.3 Theoretical Consistency Between PSDMig and Least-Squares Migration

  • Under time consistency, PSDMig's propagation-operator cancellation is equivalent to the LSM Hessian inverse's role and targets amplitude-preserved, high-resolution reflectivity without solving least squares.The paper characterizes PSDMig's deconvolution as an analytical, physically driven compensation.Source: §7.3 · p. 13

7.4 Essential Difference Between PSDMig and FWI-Imaging

  • Unlike FWI-Imaging's edge detection on an FWI result, PSDMig inverts primary reflections for local boundary reflectivity, requiring only a kinematically accurate smooth background and defining reflectivity along incident propagation.The paper says FWI-Imaging depends strongly on FWI result quality, whereas PSDMig does not require a high-resolution FWI result.Source: §7.4 · pp. 13–14

7.5 Data and Model Requirements of the Three Methods

  • PSDFWI requires broadband, wide-azimuth data; PSDSI and PSDMig require primary reflections and smooth, kinematically accurate models, with angle decomposition for PSDSI.For isotropic elastic waves, PSDSI and PSDMig do not require P/S wave decomposition under the stated model conditions; the three methods add no extra data or computational burden relative to mainstream methods.Source: §7.5 · p. 14

7.6 Previous Numerical Validation

  • Prior tests reported PSDFWI mitigation of cycle skipping and better deconvolution-type correction; PSDSI feasibility; and PSDMig gains in resolution and phase fidelity over cross-correlation imaging.The authors present these numerical validations as conceptual support for the framework and evidence that its methods are computationally feasible.Source: §7.6 · p. 14

7.7 Tension Between Theory and Engineering Implementation

  • The framework's derivation assumes time consistency and the first-order Born approximation, while real-data deconvolution stability and operator cancellation depend on noise, illumination, traveltime accuracy, acquisition, and frequency band.The deconvolution's analytical form does not remove the need for special numerical treatment with band-limited, noisy, or illumination-limited data.Source: §7.7 · p. 14

8 Conclusions

PSDWII recasts waveform inversion imaging around seismic propagation’s physical structure and virtual sources. Its shared framework supports three imaging methods, using physical operations to separate parameter effects and target distinct inversion levels.

8 Conclusions

  • PSDWII shifts waveform inversion imaging from mathematical optimization toward a physically interpretable, computationally concise physical-structure-driven paradigm.The framework does not construct objective functions or compute their gradients or Hessian matrices and inverses.Source: §8 · p. 15
  • Seismic propagation follows “two propagations, one virtual source”: incident waves interact with heterogeneities to generate radiating virtual sources, then secondary waves propagate and are received.This three-step physical structure forms the framework’s foundation.Source: §8 · p. 15
  • The framework projects adjoint wavefields as approximate virtual sources, then deconvolves to remove incident-wave and radiation-pattern effects before extracting target parameters.Timing consistency between the virtual source and incident wavefield is necessary to identify the adjoint wavefield as the true virtual source.Source: §8 · p. 15
  • Deconvolution separates incident-wave and radiation-pattern contributions from virtual sources, avoiding methodological mathematical coupling and the need for Hessian-based compensation.Traditional gradient cross-correlation instead mixes parameter contributions.Source: §8 · p. 15
  • PSDFWI, PSDSI, and PSDMig share a methodological core but use different virtual-source expressions for model building, lithologic parameter inversion, and structure-boundary imaging.Their data, initial-model, and computational requirements are comparable to mainstream methods, without extra burden.Source: §8 · p. 15
  • Physical decoupling uses differences in parameter radiation-pattern angular dependence, and its effectiveness depends importantly on model parameterization.The framework does not construct objective functions or compute gradients or Hessian matrices and inverses.Source: §8 · p. 15
  • PSDMig defines local reflectivity along the incident-wave direction, extending beyond the interface-normal special case to arbitrary incidence angles and complex structures.The paper states that PSDMig shares least-squares migration’s theoretical goal but uses deconvolution and is more computationally efficient.Source: §8 · p. 15
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