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Attenuation Compensation in Lossy Media via the Wave Operator Model

Tianchen Shao, Zekui Jia, Maokun Li, Shenheng Xu, Fan Yang

TL;DR

This work addresses inverse scattering in lossy media where attenuation degrades waveform information. It derives a non-closed-form electric-field solution within the wave-operator framework, decomposing the field into a propagation term $E^{Pro}$ and a dissipation term $E^{Dis}$, and analyzes a Green's function that is exact under uniform loss. An attenuation-compensation strategy is proposed by applying an exponential gain with an estimated $p_{app}$ based on a dissipation fraction $k(p_1,p_2)$, enabling ROM-based parameter inversion in lossy environments. Numerical 2-D FDTD experiments across uniform, linear inhomogeneous, and biased sinusoidal dissipation distributions demonstrate accurate restoration of attenuated data to near-lossless levels (per-trace $L_2$ misfits around 3–5%) and robustness, establishing a foundation for practical ROM-enabled imaging in lossy media.

Abstract

The wave operator model provides a framework for modeling wave propagation by encoding material parameter distributions into matrix-form operators. This paper extends this framework from lossless to lossy media. We present a derivation of the wave operator solution for the electric field in dissipative environments, which can be decomposed into a closed-form propagation term and a non-closed-form dissipation term. Based on an analysis of the dominant exponential decay within the propagation term, an attenuation compensation strategy is proposed to restore the attenuated data to an approximate lossless state. The performance of this compensation strategy is analyzed and validated through numerical experiments, establishing the theoretical foundation for reduced order model (ROM)-based techniques in lossy media.

Attenuation Compensation in Lossy Media via the Wave Operator Model

TL;DR

This work addresses inverse scattering in lossy media where attenuation degrades waveform information. It derives a non-closed-form electric-field solution within the wave-operator framework, decomposing the field into a propagation term and a dissipation term , and analyzes a Green's function that is exact under uniform loss. An attenuation-compensation strategy is proposed by applying an exponential gain with an estimated based on a dissipation fraction , enabling ROM-based parameter inversion in lossy environments. Numerical 2-D FDTD experiments across uniform, linear inhomogeneous, and biased sinusoidal dissipation distributions demonstrate accurate restoration of attenuated data to near-lossless levels (per-trace misfits around 3–5%) and robustness, establishing a foundation for practical ROM-enabled imaging in lossy media.

Abstract

The wave operator model provides a framework for modeling wave propagation by encoding material parameter distributions into matrix-form operators. This paper extends this framework from lossless to lossy media. We present a derivation of the wave operator solution for the electric field in dissipative environments, which can be decomposed into a closed-form propagation term and a non-closed-form dissipation term. Based on an analysis of the dominant exponential decay within the propagation term, an attenuation compensation strategy is proposed to restore the attenuated data to an approximate lossless state. The performance of this compensation strategy is analyzed and validated through numerical experiments, establishing the theoretical foundation for reduced order model (ROM)-based techniques in lossy media.

Paper Structure

This paper contains 13 sections, 3 theorems, 60 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Wave Operator Solution
  4. Attenuation Compensation
  5. Numerical Experiments
  6. Case 1: Uniform Dissipation Distribution
  7. Case 2: Linear Inhomogeneous Dissipation
  8. Case 3: Biased Sinusoidal Inhomogeneous Dissipation
  9. Boundary of Compensation Method
  10. Conclusion
  11. Derivation of the Green's Function
  12. Approximation of the Dissipation Fraction
  13. Proof of Error Bound of the Dissipation Fraction

Key Result

Proposition 1

The solution to the Green's function in Wave Equation 1 is determined by the spatial distribution of the dissipation parameter $p(\boldsymbol{x})$. 1) Uniform Dissipation: For a medium with a spatially uniform dissipation parameter $p(\boldsymbol{x})=p$, where $p$ is a constant (i.e., $\sigma(\bolds where $A'=A-p^2/4$ and $H(t)$ is the Heaviside step function. 2) Non-Uniform Dissipation: In a medi

Figures (12)

  • Figure 1: Contrast distribution of the three-stripe target example. Yellow stripes represent the target on a homogeneous background, with sensors marked by red crosses. The colorbar indicates the contrast value.
  • Figure 2: Comparison between $y=k(p, 1)$ and $y=p$.
  • Figure 3: Comparison of lossless data, measured data and compensated data with uniform dissipation distribution. The transmitter is at Sensor 1 and receivers are at (a) Sensor 5, (b) Sensor 10, (c) Sensor 15, (d) Sensor 20.
  • Figure 4: Common shot gathers at Sensor 10 with uniform dissipation distribution. The colorbar indicates the magnitude of the collected data. (a) Synthetic data in lossless environment. (b) Measured data in lossy environment. (c) Compensated data.
  • Figure 5: (a) Distribution of dissipation parameter $p(\boldsymbol{x})$. (b) Distribution of conductivity $\sigma(\boldsymbol{x})$. (c) Conductivity profile along the vertical line at $\boldsymbol{x}=60$ m.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Lemma 1
  • proof