We consider the reconstruction of a spatially-dependent scattering co- ecient in a linear transpo... more We consider the reconstruction of a spatially-dependent scattering co- ecient in a linear transport equation from diusion-type measurements. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an addi- tional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We
We give a detailed mathematical analysis of the radiative transport limit for the average phase s... more We give a detailed mathematical analysis of the radiative transport limit for the average phase space density of solutions of the Schroedinger equation with time dependent random potential. Our derivation is based on the construction of an approximate martingale for the random Wigner distribution.
We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation... more We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. This allows us to construct an approximate martingale for the phase space Wigner transform of two wave fields. Using a priori $L^2$-bounds available in the time-reversal setting, we prove that the Wigner transform
ABSTRACT We describe a method for accelerating a 3D Monte Carlo forward radiative transfer model ... more ABSTRACT We describe a method for accelerating a 3D Monte Carlo forward radiative transfer model to the point where it can be used in a new kind of Bayesian retrieval framework. The remote sensing challenge is to detect and quantify a chemical effluent of a known absorbing gas produced by an industrial facility in a deep valley. The available data is a single low-resolution noisy image of the scene in the near IR at an absorbing wavelength for the gas of interest. The detected sunlight has been multiply reflected by the variable terrain and/or scattered by an aerosol that is assumed partially known and partially unknown. We thus introduce a new class of remote sensing algorithms best described as “multi-pixel” techniques that call necessarily for a 3D radiative transfer model (but demonstrated here in 2D); they can be added to conventional ones that exploit typically multi-or hyper-spectral data, sometimes with multi-angle capability, with or without information about polarization. The novel Bayesian inference methodology uses adaptively, with efficiency in mind, the fact that a Monte Carlo forward model has a known and controllable uncertainty depending on the number of sun-to-detector paths used.
ABSTRACT We consider scattering of a pulse propagating through a three-dimensional random media a... more ABSTRACT We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty–Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully three-dimensional situation.
The full application of Bayesian inference to inverse problems requires exploration of a posterio... more The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called M C 3 that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved.
We study a one-dimensional elliptic problem with highly oscillatory random diffusion coefficient.... more We study a one-dimensional elliptic problem with highly oscillatory random diffusion coefficient. We derive a homogenized solution and a so-called Gaussian corrector. We also prove a "pointwise" large deviation principle (LDP) for the full solution and approximate this LDP with a more tractable form. Applications to uncertainty quantification are considered.
Radiative transfer theory provides a good framework for the study of multiple scattering in the r... more Radiative transfer theory provides a good framework for the study of multiple scattering in the randomly inhomogeneous lithosphere. Envelopes of high-frequency seismograms (mainly S coda waves) of local earthquakes have been synthesized on the basis of this theory, and inversions for some Earth parameters such as intrinsic attenuation, scattering attenuation and degree of non-isotropic scattering have been carried out. However, a scalar model has often been assumed because of its mathematical relative simplicity. The simpli¢cation amounts to neglecting the polarized nature of the underlying motion. This approach is only valid for long lapse times when S waves become unpolarized because of high-order scattering, and cannot be justi¢ed by only assuming that the source is unpolarized. We show that incoming unpolarized S waves can be up to 80 per cent polarized after single scattering. Depolarization of S waves after multiple scattering is studied by a Monte Carlo method. We show that the scattering of S waves o¡ di¡erent kinds of inhomogeneities gives rise to di¡erent polarization and depolarization patterns. Consequently, polarization should provide valuable information for the understanding of the physics of wave motion and the properties of the Earth's lithosphere.
We study transport for a scalar model of Love waves. These waves arise in the propagation of seis... more We study transport for a scalar model of Love waves. These waves arise in the propagation of seismic waves whose energy is concentrated in the vicinity of the earth surface. We derive radiative transfer equations from first principles for the angularly resolved energy density of the Love waves in a simplified acoustic model. We consider a rough top surface with weak fluctuations at the scale of the wavelength. The transport equation accounts for the multiple scattering of the Love waves and their scattering into volume waves. We also analyze a diffusive regime when energy is universally distributed over various modes of the Love waves.
We consider the matrix-valued radiative transfer equations for the Stokes parameters for the prop... more We consider the matrix-valued radiative transfer equations for the Stokes parameters for the propagation of light through turbulent atmospheres. A Monte Carlo method is introduced to solve the time dependent matrix-valued radiative transfer equations in 3D geometry. The Monte Carlo method is based on a probabilistic representation of the radiative transfer equations involving an augmented scalar transport equation where the polarization parameters are independent variables. The linear moments of the augmented transport equation with respect to the polarization parameters solve the matrix-valued radiative transfer equations. We show how polarization and depolarization e!ects develop in time for isotropic and unpolarized point sources, considered for concreteness in spherical and half-space geometries. We analyze in detail the creation of polarization by single-and multiple-scattering e!ects. : S 0 0 2 2 -4 0 7 3 ( 0 0 ) 0 0 1 2 4 -2 (r, t)"
Transport theoretic boundary conditions are derived for acoustic wave reflection and transmission... more Transport theoretic boundary conditions are derived for acoustic wave reflection and transmission at a rough interface with small random fluctuations. The Wigner distribution is used to go from waves to energy transport in the high frequency limit, and the Born expansion is used to calculate the effect of the random rough surface. The smoothing method is also used to remove
ABSTRACT Remote sensing is a key tool for sorting cloud ensembles by dynamical state, aerosol env... more ABSTRACT Remote sensing is a key tool for sorting cloud ensembles by dynamical state, aerosol environments by source region, and establishing causal relationships between aerosol amounts, type, and cloud microphysics-the so-called indirect aerosol climate impacts, and one of the main sources of uncertainty in current climate models. Current satellite imagers use data processing approaches that invariably start with cloud detection/masking to isolate aerosol air-masses from clouds, and then rely on one-dimensional (1D) radiative transfer (RT) to interpret the aerosol and cloud measurements in isolation. Not only does this lead to well-documented biases for the estimates of aerosol radiative forcing and cloud optical depths in current missions, but it is fundamentally inadequate for future missions such as EarthCARE where capturing the complex, three-dimensional (3D) interactions between clouds and aerosols is a primary objective. In order to advance the state of the art, the next generation of satellite information processing systems must incorporate technologies that will enable the treatment of the atmosphere as a fully 3D environment, represented more realistically as a continuum. At one end, there is an optically thin background dominated by aerosols and molecular scattering that is strongly stratified and relatively homogeneous in the horizontal. At the other end, there are optically thick embedded elements, clouds and aerosol plumes, which can be more or less uniform and quasi-planar or else highly 3D with boundaries in all directions; in both cases, strong internal variability may be present. To make this paradigm shift possible, we propose to combine the standard models for satellite signal prediction physically grounded in 1D and 3D RT, both scalar and vector, with technologies adapted from biomedical imaging, digital image processing, and computer vision. This will enable us to demonstrate how the 3D distribution of atmospheric constituents, and their associated microphysical properties, can be reconstructed from multi-angle/multi-spectral imaging radiometry and, more and more, polarimetry. Specific technologies of interest are computed tomography (reconstruction from projections), optical tomography (using cross-pixel radiation transport in the diffusion limit), stereoscopy (depth/height retrievals), blind source and scale separation (signal unmixing), and disocclusion (information recovery in the presence of obstructions). Later on, these potentially powerful inverse problem solutions will be fully integrated in a versatile satellite data analysis toolbox. At present, we can report substantial progress at the component level. Specifically, we will focus on the most elementary problems in atmospheric tomography with an emphasis on the vastly under-exploited class of multi-pixel techniques. One basic problem is to infer the outer shape and mean opacity of 3D clouds, along with a bulk measure of cloud particle size. Another is to separate high and low cloud layers based on their characteristically different spatial textures. Yet another is to reconstruct the 3D spatial distribution of aerosol density based on passive imaging. This suite of independent feasibility studies amounts to a compelling proofof- concept for the ambitious 3D-Tomographic Reconstruction of the Aerosol-Cloud Environment (3D-TRACE) project as a whole.
This paper concerns the homogenization problem of a parabolic equation with large, timedependent,... more This paper concerns the homogenization problem of a parabolic equation with large, timedependent, random potentials in high dimensions d ≥ 3. Depending on the competition between temporal and spatial mixing of the randomness, the homogenization procedure turns to be different. We characterize the difference by proving the corresponding weak convergence of Brownian motion in random scenery. When the potential depends on the spatial variable macroscopically, we prove a convergence to SPDE.
We consider the reconstruction of a spatially-dependent scattering co- ecient in a linear transpo... more We consider the reconstruction of a spatially-dependent scattering co- ecient in a linear transport equation from diusion-type measurements. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an addi- tional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We
We give a detailed mathematical analysis of the radiative transport limit for the average phase s... more We give a detailed mathematical analysis of the radiative transport limit for the average phase space density of solutions of the Schroedinger equation with time dependent random potential. Our derivation is based on the construction of an approximate martingale for the random Wigner distribution.
We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation... more We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. This allows us to construct an approximate martingale for the phase space Wigner transform of two wave fields. Using a priori $L^2$-bounds available in the time-reversal setting, we prove that the Wigner transform
ABSTRACT We describe a method for accelerating a 3D Monte Carlo forward radiative transfer model ... more ABSTRACT We describe a method for accelerating a 3D Monte Carlo forward radiative transfer model to the point where it can be used in a new kind of Bayesian retrieval framework. The remote sensing challenge is to detect and quantify a chemical effluent of a known absorbing gas produced by an industrial facility in a deep valley. The available data is a single low-resolution noisy image of the scene in the near IR at an absorbing wavelength for the gas of interest. The detected sunlight has been multiply reflected by the variable terrain and/or scattered by an aerosol that is assumed partially known and partially unknown. We thus introduce a new class of remote sensing algorithms best described as “multi-pixel” techniques that call necessarily for a 3D radiative transfer model (but demonstrated here in 2D); they can be added to conventional ones that exploit typically multi-or hyper-spectral data, sometimes with multi-angle capability, with or without information about polarization. The novel Bayesian inference methodology uses adaptively, with efficiency in mind, the fact that a Monte Carlo forward model has a known and controllable uncertainty depending on the number of sun-to-detector paths used.
ABSTRACT We consider scattering of a pulse propagating through a three-dimensional random media a... more ABSTRACT We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty–Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully three-dimensional situation.
The full application of Bayesian inference to inverse problems requires exploration of a posterio... more The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called M C 3 that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved.
We study a one-dimensional elliptic problem with highly oscillatory random diffusion coefficient.... more We study a one-dimensional elliptic problem with highly oscillatory random diffusion coefficient. We derive a homogenized solution and a so-called Gaussian corrector. We also prove a "pointwise" large deviation principle (LDP) for the full solution and approximate this LDP with a more tractable form. Applications to uncertainty quantification are considered.
Radiative transfer theory provides a good framework for the study of multiple scattering in the r... more Radiative transfer theory provides a good framework for the study of multiple scattering in the randomly inhomogeneous lithosphere. Envelopes of high-frequency seismograms (mainly S coda waves) of local earthquakes have been synthesized on the basis of this theory, and inversions for some Earth parameters such as intrinsic attenuation, scattering attenuation and degree of non-isotropic scattering have been carried out. However, a scalar model has often been assumed because of its mathematical relative simplicity. The simpli¢cation amounts to neglecting the polarized nature of the underlying motion. This approach is only valid for long lapse times when S waves become unpolarized because of high-order scattering, and cannot be justi¢ed by only assuming that the source is unpolarized. We show that incoming unpolarized S waves can be up to 80 per cent polarized after single scattering. Depolarization of S waves after multiple scattering is studied by a Monte Carlo method. We show that the scattering of S waves o¡ di¡erent kinds of inhomogeneities gives rise to di¡erent polarization and depolarization patterns. Consequently, polarization should provide valuable information for the understanding of the physics of wave motion and the properties of the Earth's lithosphere.
We study transport for a scalar model of Love waves. These waves arise in the propagation of seis... more We study transport for a scalar model of Love waves. These waves arise in the propagation of seismic waves whose energy is concentrated in the vicinity of the earth surface. We derive radiative transfer equations from first principles for the angularly resolved energy density of the Love waves in a simplified acoustic model. We consider a rough top surface with weak fluctuations at the scale of the wavelength. The transport equation accounts for the multiple scattering of the Love waves and their scattering into volume waves. We also analyze a diffusive regime when energy is universally distributed over various modes of the Love waves.
We consider the matrix-valued radiative transfer equations for the Stokes parameters for the prop... more We consider the matrix-valued radiative transfer equations for the Stokes parameters for the propagation of light through turbulent atmospheres. A Monte Carlo method is introduced to solve the time dependent matrix-valued radiative transfer equations in 3D geometry. The Monte Carlo method is based on a probabilistic representation of the radiative transfer equations involving an augmented scalar transport equation where the polarization parameters are independent variables. The linear moments of the augmented transport equation with respect to the polarization parameters solve the matrix-valued radiative transfer equations. We show how polarization and depolarization e!ects develop in time for isotropic and unpolarized point sources, considered for concreteness in spherical and half-space geometries. We analyze in detail the creation of polarization by single-and multiple-scattering e!ects. : S 0 0 2 2 -4 0 7 3 ( 0 0 ) 0 0 1 2 4 -2 (r, t)"
Transport theoretic boundary conditions are derived for acoustic wave reflection and transmission... more Transport theoretic boundary conditions are derived for acoustic wave reflection and transmission at a rough interface with small random fluctuations. The Wigner distribution is used to go from waves to energy transport in the high frequency limit, and the Born expansion is used to calculate the effect of the random rough surface. The smoothing method is also used to remove
ABSTRACT Remote sensing is a key tool for sorting cloud ensembles by dynamical state, aerosol env... more ABSTRACT Remote sensing is a key tool for sorting cloud ensembles by dynamical state, aerosol environments by source region, and establishing causal relationships between aerosol amounts, type, and cloud microphysics-the so-called indirect aerosol climate impacts, and one of the main sources of uncertainty in current climate models. Current satellite imagers use data processing approaches that invariably start with cloud detection/masking to isolate aerosol air-masses from clouds, and then rely on one-dimensional (1D) radiative transfer (RT) to interpret the aerosol and cloud measurements in isolation. Not only does this lead to well-documented biases for the estimates of aerosol radiative forcing and cloud optical depths in current missions, but it is fundamentally inadequate for future missions such as EarthCARE where capturing the complex, three-dimensional (3D) interactions between clouds and aerosols is a primary objective. In order to advance the state of the art, the next generation of satellite information processing systems must incorporate technologies that will enable the treatment of the atmosphere as a fully 3D environment, represented more realistically as a continuum. At one end, there is an optically thin background dominated by aerosols and molecular scattering that is strongly stratified and relatively homogeneous in the horizontal. At the other end, there are optically thick embedded elements, clouds and aerosol plumes, which can be more or less uniform and quasi-planar or else highly 3D with boundaries in all directions; in both cases, strong internal variability may be present. To make this paradigm shift possible, we propose to combine the standard models for satellite signal prediction physically grounded in 1D and 3D RT, both scalar and vector, with technologies adapted from biomedical imaging, digital image processing, and computer vision. This will enable us to demonstrate how the 3D distribution of atmospheric constituents, and their associated microphysical properties, can be reconstructed from multi-angle/multi-spectral imaging radiometry and, more and more, polarimetry. Specific technologies of interest are computed tomography (reconstruction from projections), optical tomography (using cross-pixel radiation transport in the diffusion limit), stereoscopy (depth/height retrievals), blind source and scale separation (signal unmixing), and disocclusion (information recovery in the presence of obstructions). Later on, these potentially powerful inverse problem solutions will be fully integrated in a versatile satellite data analysis toolbox. At present, we can report substantial progress at the component level. Specifically, we will focus on the most elementary problems in atmospheric tomography with an emphasis on the vastly under-exploited class of multi-pixel techniques. One basic problem is to infer the outer shape and mean opacity of 3D clouds, along with a bulk measure of cloud particle size. Another is to separate high and low cloud layers based on their characteristically different spatial textures. Yet another is to reconstruct the 3D spatial distribution of aerosol density based on passive imaging. This suite of independent feasibility studies amounts to a compelling proofof- concept for the ambitious 3D-Tomographic Reconstruction of the Aerosol-Cloud Environment (3D-TRACE) project as a whole.
This paper concerns the homogenization problem of a parabolic equation with large, timedependent,... more This paper concerns the homogenization problem of a parabolic equation with large, timedependent, random potentials in high dimensions d ≥ 3. Depending on the competition between temporal and spatial mixing of the randomness, the homogenization procedure turns to be different. We characterize the difference by proving the corresponding weak convergence of Brownian motion in random scenery. When the potential depends on the spatial variable macroscopically, we prove a convergence to SPDE.
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Papers by Guillaume Bal