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Sediment Concentration Estimation via Multiscale Inverse Problem and Stochastic Homogenization

Jiwei Li, Lingyun Qiu, Zhongjing Wang, Hui Yu, Siqin Zheng

TL;DR

The paper addresses estimating macroscopic sediment concentration in a multiscale, sediment-laden flow from partial boundary measurements. It adopts a multiscale inverse problem framework grounded in stochastic homogenization to derive an effective medium model with $m(x)=\frac{p(x)}{c_1^2}+\frac{1-p(x)}{c_0^2}$ and the time-domain equation $(m(x)\partial_t^2-\Delta)u=f$, enabling efficient inversion on coarse meshes. It contributes rigorous homogenization results with error bounds, gradient-based inversion using $L^2$ and $W^2$ objectives, and stabilization techniques such as mollification and shot averaging, demonstrated through 2D numerical experiments that recover sediment concentration with high fidelity. The method offers a computationally efficient pathway for boundary-data-based sediment-concentration estimation in complex environments, with potential extensions to higher-order homogenization and broader stochastic media analyses.

Abstract

In this work, we contribute to the broader understanding of inverse problems by introducing a versatile multiscale modeling framework tailored to the challenges of sediment concentration estimation. Specifically, we propose a novel approach for sediment concentration measurement in water flow, modeled as a multiscale inverse medium problem. To address the multiscale nature of the sediment distribution, we treat it as an inhomogeneous random field and use the homogenization theory in deriving the effective medium model. The inverse problem is formulated as the reconstruction of the effective medium model, specifically, the sediment concentration, from partial boundary measurements. Additionally, we develop numerical algorithms to improve the efficiency and accuracy of solving this inverse problem. Our numerical experiments demonstrate the effectiveness of the proposed model and methods in producing accurate sediment concentration estimates, offering new insights into sediment concentration measurement in complex environments.

Sediment Concentration Estimation via Multiscale Inverse Problem and Stochastic Homogenization

TL;DR

The paper addresses estimating macroscopic sediment concentration in a multiscale, sediment-laden flow from partial boundary measurements. It adopts a multiscale inverse problem framework grounded in stochastic homogenization to derive an effective medium model with and the time-domain equation , enabling efficient inversion on coarse meshes. It contributes rigorous homogenization results with error bounds, gradient-based inversion using and objectives, and stabilization techniques such as mollification and shot averaging, demonstrated through 2D numerical experiments that recover sediment concentration with high fidelity. The method offers a computationally efficient pathway for boundary-data-based sediment-concentration estimation in complex environments, with potential extensions to higher-order homogenization and broader stochastic media analyses.

Abstract

In this work, we contribute to the broader understanding of inverse problems by introducing a versatile multiscale modeling framework tailored to the challenges of sediment concentration estimation. Specifically, we propose a novel approach for sediment concentration measurement in water flow, modeled as a multiscale inverse medium problem. To address the multiscale nature of the sediment distribution, we treat it as an inhomogeneous random field and use the homogenization theory in deriving the effective medium model. The inverse problem is formulated as the reconstruction of the effective medium model, specifically, the sediment concentration, from partial boundary measurements. Additionally, we develop numerical algorithms to improve the efficiency and accuracy of solving this inverse problem. Our numerical experiments demonstrate the effectiveness of the proposed model and methods in producing accurate sediment concentration estimates, offering new insights into sediment concentration measurement in complex environments.

Paper Structure

This paper contains 20 sections, 3 theorems, 61 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Review of sediment concentration measurement methods
  3. Mathematical model
  4. Acoustic structure of sediment-laden flow
  5. Wave propagation in heterogeneous medium
  6. Stochastic homogenization of wave equations
  7. Proof of \ref{['thm_homogenization']}
  8. Multiscale inverse problem
  9. Numerical approach
  10. Closed form of effective models
  11. Methods to solve the inverse problems
  12. Determination of sediment concentration
  13. Numerical results
  14. Verification of the shot data containing information of sediment concentration
  15. Data comparison
  16. ...and 5 more sections

Key Result

Theorem 1

If $\rho\in C^\alpha(\mathbb{R}^d)$ for some $\alpha\in(0,1]$ with the $\alpha$-Hölder seminorm and support $\bar{D}$, then the homogenized equation of eq:hete is where Moreover, there exists a constant $C$, depending on $k,\underline{n}^2_0,\underline{n}^2_1,d,\alpha,[\rho]_\alpha,|D|$, such that

Figures (7)

  • Figure 1: The diagram of numerical simulations.
  • Figure 2: The sediment concentration. (a) Sediment sample $c_\epsilon(x)$ from Gaussian function. (b) Sediment sample $c_\epsilon(x)$ from chiuMathematicalModelsDistribution2000. (c) Effective medium model $c(x)$ from Gaussian function. (d) Effective medium model $c(x)$ from chiuMathematicalModelsDistribution2000.
  • Figure 3: Comparison of shot data in the sediment-laden and sediment-free media. (a) Shot data in the sediment-free media. (b) Shot data in the sediment-laden model \ref{['eq_gaussian']}. (c) Shot data in the sediment-laden model \ref{['eq_chiu']}. (d) Comparison of the shot data in the sediment-laden and sediment-free media.
  • Figure 4: The measurements of the wave field in the heterogeneous medium and the effective medium. The top row shows the results for the Gaussian function, while the bottom row shows the results for the sediment concentration from chiuMathematicalModelsDistribution2000. (a) Effective data $d_{\rm eff}$. (b) Heterogeneous data $d_{\rm het}$. (c) Averaged heterogeneous data. (d) Comparison for the data from the Gaussian function. (e) Effective data $d_{\rm eff}$. (f) Heterogeneous data $d_{\rm het}$. (g) Averaged heterogeneous data. (h) Comparison for the data from chiuMathematicalModelsDistribution2000.
  • Figure 5: The true model, and inverted models with $L^2$ and $W^2$ objective function. The top row shows the results for the Gaussian function, while the bottom row shows the results for the sediment concentration from chiuMathematicalModelsDistribution2000. (a) True effective model. (b) Inverted model with $L^2$ objective function. (c) The inverted model with $W^2$ objective function. (d) True effective model. (e) Inverted model with $L^2$ objective function. (f) The inverted model with $W^2$ objective function.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of \ref{['thm_homogenization']}
  • Remark 1
  • Remark 2