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This paper addresses the challenging and interesting inverse problem of reconstructing the spatially varying dielectric constant of a medium from phaseless backscattering measurements generated by single-point illumination. The underlying mathematical model is governed by the three-dimensional Helmholtz equation, and the available data consist solely of the magnitude of the scattered wave field. To address the nonlinearity and servere ill-posedness of this phaseless inverse scattering problem, we introduce a robust, globally convergent numerical framework combining several key regularization strategies. Our method first employs a phase retrieval step based on the Wentzel--Kramers--Brillouin (WKB) ansatz, where the lost phase information is reconstructed by solving a nonlinear optimization problem. Subsequently, we implement a Fourier-based dimension reduction technique, transforming the original problem into a more stable system of elliptic equations with Cauchy boundary conditions. To solve this resulting system reliably, we apply the Carleman convexification approach, constructing a strictly convex weighted cost functional whose global minimizer provides an accurate approximation of the true solution. Numerical simulations using synthetic data with high noise levels demonstrate the effectiveness and robustness of the proposed method, confirming its capability to accurately recover both the geometric location and contrast of hidden scatterers.
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A data-driven approach for fast atmospheric radiative transfer inversion
Abstract Far-infrared Outgoing Radiation Understanding and Monitoring (FORUM) was selected in 2019 as the ninth Earth Explorer mission by the European Space Agency. Its primary objective is to collect interferometric measurements in the far-infrared (FIR) spectral range, which accounts for 50% of Earth’s outgoing longwave radiation emitted into space, and will be observed from space for the first time. Accurate measurements of the FIR at the top of the atmosphere are crucial for improving climate models. Current instruments are insufficient, necessitating the development of advanced computational techniques. FORUM will provide unprecedented insights into key atmospheric parameters, such as surface emissivity, water vapor, and ice cloud properties, through the use of a Fourier transform spectrometer. To ensure the quality of the mission’s data, an end-to-end simulator was developed to simulate the measurement process and evaluate the effects of instrument characteristics and environmental factors. The core challenge of the mission is solving the retrieval problem, which involves estimating atmospheric properties from the radiance spectra observed by the satellite. This problem is ill-posed and regularization techniques are necessary to stabilize the solution. In this work, we present a data-driven approach to approximate the inverse mapping in the retrieval problem, aiming to achieve a solution that is both computationally efficient and accurate. In the first phase, we generate an initial approximation of the inverse mapping using only simulated FORUM data. In the second phase, we improve this approximation by introducing climatological data asa prioriinformation and using a neural network to estimate the optimal regularization parameters during the retrieval process. While our approach does not match the precision of full-physics retrieval methods, its key advantage is the ability to deliver results almost instantaneously, making it highly suitable for real-time applications. Furthermore, the proposed method can provide more accuratea prioriestimates for full-physics methods, thereby improving the overall accuracy of the retrieved atmospheric profiles.
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- Award ID(s):
- 2152661
- PAR ID:
- 10625590
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 41
- Issue:
- 8
- ISSN:
- 0266-5611
- Format(s):
- Medium: X Size: Article No. 085006
- Size(s):
- Article No. 085006
- Sponsoring Org:
- National Science Foundation
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