Cloud and aerosol properties retrieved from the inversion of remote sensing measurements (Stephens and Kummerow, 2007; Dubovik et al., 2011) are a critical source of information for understanding and testing the closure of the Earth's radiation budget (Raschke et al., 2005; McFarlane et al., 2016; Zhou et al., 2016), validating dynamical atmospheric models of varying complexity (Hack et al., 2006; Endo et al., 2015; Bodas-Salcedo et al., 2016), and developing parameterizations in large-scale models (Hill et al., 2012; Xie and Zhang, 2015). Cloud radiative feedbacks and aerosol–cloud interactions (ACIs; Bellouin et al., 2020), particularly in cumuliform clouds (Sherwood et al., 2014; Vial et al., 2018), are key sources of uncertainty in projections of future climate (Sherwood et al., 2020) and the forecasting of weather (Van Weverberg et al., 2018) and solar energy (Jimenez et al., 2016).

As dynamical modeling of the atmosphere and climate becomes more and more complex, there is a greater demand for high-quality observations to constrain the uncertain processes within the models and inform model development (Morrison et al., 2020). New observational techniques are required that can provide robust statistics of small-scale, spatially resolved cloud and aerosol microphysical parameters so that their controlling processes can be constrained in both high- and low-resolution modeling. We describe a novel remote sensing retrieval technique with the potential to meet these needs by providing 3D instantaneous snapshots of volumetric properties of the atmosphere, thus making complete use of the resolution of the sensors.

Scattered solar radiation is one of the best candidates for providing high-resolution constraints on aerosol and cloud microphysics, due to its well-documented sensitivity to the properties of particles in those size ranges (Dubovik et al., 2002; King and Vaughan, 2012; Dzambo et al., 2021; Ewald et al., 2021) and our ability to design narrowband sensors that can cost-effectively reach a high spatial resolution (tens of meters) from space. Typical cloud and aerosol remote sensing retrieval algorithms do not utilize realistic 3D radiative transfer (RT) models to interpret measured scattered solar radiation (e.g., Dubovik et al., 2011; Grosvenor et al., 2018). Instead, they make use of two key simplifying assumptions when interpreting radiance measurements. First, they assume that the media (e.g., clouds) form horizontally homogeneous, plane-parallel layers within the field of view of each radiance measurement. Second, they assume that there is no radiative interaction between the regions within the field of view of each radiance measurement in what is known as the independent pixel approximation (IPA).

These assumptions dramatically reduce the computational complexity of the retrieval process, but they also compromise retrieval accuracy (Marshak et al., 2006; Zhang et al., 2012; Kato and Marshak, 2009), leading to errors in, for example, cloud optical depth that can have domain biases of ∼35 % in cumuliform clouds (Seethala, 2012). These errors contribute to the large inter-instrument inconsistencies in retrievals (Di Girolamo et al., 2010; Lebsock and Su, 2014; Ahn et al., 2018; Fu et al., 2019; Painemal et al., 2021). These errors arise because clouds and aerosols can have strong horizontal gradients in their physical and optical properties (Marshak et al., 1997; Gerber et al., 2001; Zhao and Di Girolamo, 2007; Kahn et al., 2007), which breaks the assumption of the IPA and necessitates the use of 3D RT to accurately model the transport of radiation (Davies, 1978; Cahalan et al., 2005).

These errors distort into our climate records and also affect retrievals made using a combination of passive and active sensors (Saito et al., 2019). The retrieval assumptions also limit the amount of detail about the cloud and aerosol fields that can be retrieved using passive sensing as, once the IPA is imposed, there is no way to identify the vertical geometric variability in cloud microphysics within a layer without active instruments, which in the case of cloud radar requires strong assumptions about the shape of the particle size distribution. An algorithmic advance in operational retrievals is required to better extract the information about 3D variability in cloud and aerosol microphysics contained within high-resolution solar radiance measurements to provide the novel observations required for advancing cloud and aerosol science.

Many algorithms have been demonstrated that utilize 3D RT to improve remote sensing retrievals of cloud properties but have been limited to using radiometric information from a single mono-angle imager at a time (Marshak et al., 1998a; Marchand and Ackerman, 2004; Zinner et al., 2006; Cornet and Davies, 2008) or a single zenith radiance measurement (Fielding et al., 2014). This restriction limits the amount of information obtainable from the radiance field, so that only column integrals or horizontal averages of cloud properties can be inferred, rather than their three-dimensional variability. As a result, several of these methods have relied on external sources of information, such as scanning radar or in situ data (Marchand and Ackerman, 2004; Fielding et al., 2014), or strong assumptions that are not generally applicable (Marshak et al., 1998a; Zinner et al., 2006; Cornet and Davies, 2008). It is clear that a new source of information is required to relax the strong assumptions required by these retrieval algorithms to enable the retrieval of the 3D spatial structure of cloud and aerosol microphysics.

Multi-angle imagery is a promising source of information to constrain the 3D structure of the atmosphere. Inverse problems, where multi-angle boundary measurements are used to infer internal structure, are commonly known as tomography. In atmospheric science, tomographic methods have been applied to retrieve cloud properties using non-scattering, multi-angle microwave emission (Huang et al., 2008, 2010); water vapor, using microwave attenuation (Jiang et al., 2022); and aerosol, using scattering measurements (Garay et al., 2016; Zawada et al., 2017, 2018). Multi-angle imaging has been utilized to systematically retrieve the geometric properties of clouds (Muller et al., 2002, 2007) and aerosols (Kahn et al., 2007) using stereoscopic methods.

In the field of medical imaging, multiple detectors are routinely utilized to retrieve spatially varying optical properties of the human body from multiply scattered near-infrared radiation in a process known as diffuse optical tomography (Arridge and Schotland, 2009; Bal, 2009). Taking this as inspiration, a similar tomographic approach has been proposed and formalized for the retrieval of spatially varying atmospheric constituents from multi-angle imagery of multiply scattered solar radiation in the atmospheric context (Martin et al., 2014). These methods must make use of 3D RT models, as 1D RT models are unable to reproduce the angular variations in the observed radiation field (Di Girolamo et al., 2010). Tomographic methods have the potential to provide remote sensing retrievals of volumetric cloud and aerosol properties, such as the 3D distribution of volume extinction coefficient, and possibly even microphysical quantities.

Tomography problems are commonly solved using iterative, physics-based optimization procedures similar to state-of-the-art methods in aerosol remote sensing (Xu et al., 2019; Gao et al., 2021). They can also be solved using statistical methods (Zhang and Zhang, 2019; Ronen et al., 2022) or heuristic methods, which have been explored recently in the atmospheric science context (Alexandrov et al., 2021). Tomographic methods in diffuse optical tomography typically make use of computationally efficient forward-adjoint methods to linearize a 3D RT model and calculate the cost function gradients (Arridge and Schotland, 2009). Similar methods have been employed in plane-parallel retrievals of aerosol properties (Hasekamp and Landgraf, 2005). So far, the tomographic retrievals utilizing forward-adjoint methods have only been demonstrated in atmospheric sciences in 2D (Martin and Hasekamp, 2018).

Similar tools have been developed elsewhere. The Monte Carlo 3D RT equation (RTE) solver McArtim (Deutschmann et al., 2011) is specialized for radiance derivative calculations using forward-adjoint methods but uses a backward Monte Carlo technique, similar to other work (Loeub et al., 2020), with importance sampling, which will not scale well to the multi-angle imagery required for tomography. Other Monte Carlo techniques for derivative calculations typically use forward methods with path recycling (Langmore et al., 2013; Yao et al., 2018; Czerninski and Schechner, 2021); however, most implementations of such models lack the variance reduction methods required for the efficient modeling of radiances with sharply peaked phase functions (Buras and Mayer, 2011; Wang et al., 2017) and have not been benchmarked on atmospheric problems.

Of the available 3D RTE solvers that are benchmarked on atmospheric scattering problems (Cahalan et al., 2005), the deterministic (i.e., explicit) spherical harmonics discrete ordinates method (SHDOM; Evans, 1998) is the most computationally efficient for tomography. This is due to the need to simulate many radiometric quantities. SHDOM is almost 2 orders of magnitude more computationally efficient than Monte Carlo on CPU for multi-angle imagery (Pincus and Evans, 2009). Monte Carlo solvers specialized for 3D atmospheric scattering problems have been slow to adopt a GPU-based computation, which is anticipated to give a reduction in the wall time of between 1 and 2 orders of magnitude (Efremenko et al., 2014; Ramon et al., 2019; Wang et al., 2021; Lee et al., 2022), thereby making Monte Carlo competitive against SHDOM in the future.

At the time of writing, there is no publicly available adjoint to a deterministic (i.e., explicit) 3D RTE solver appropriate to the atmospheric context like SHDOM. A forward-adjoint linearization of the SHDOM method has been developed (Doicu and Efremenko, 2019), and an SHDOM solver has been extended so that general adjoints appropriate for tomography can be computed (Doicu et al., 2022b). This forward-adjoint linearization, following the theory of Martin et al. (2014), is also technically able to compute the Jacobian matrix of partial derivatives of the forward model. However, this is computationally inefficient for tomography problems where the number of measurements is very large (Martin et al., 2014). Unfortunately, the software implementing the forward-adjoint linearization of SHDOM is not publicly available, which means we are unable to build upon these advances. Fortunately, a computationally efficient approximation to the adjoint of SHDOM has been developed and used to demonstrate the success of fully 3D retrievals of the volume extinction coefficient of clouds using multi-angle, mono-spectral imagery in a first for atmospheric remote sensing (Levis et al., 2015). This method of approximate linearization has been extended to utilize multi-spectral (Levis et al., 2017) and polarized (Levis et al., 2020) observations. This approximate linearization opens up computationally efficient access to an approximate Jacobian matrix. However, so far, only gradient-based optimization methods have been used, and it is unclear how robust or efficient the approximate linearization will be when combined with optimization methods which make direct use of the Jacobian matrix. Interestingly, the forward-adjoint method of cloud tomography using SHDOM suffered from slow convergence, and the authors only found success in their synthetic tomographic retrievals when utilizing the approximate linearization of Levis et al. (2020) and Doicu et al. (2022a) in combination with their adjoint method (Doicu et al., 2022b).

The method of Levis et al. (2020) is the most mature and successful remote sensing retrieval using solar radiances and 3D RT available in the atmospheric sciences. The method is still restricted in that its implementation is limited to isolated 3D domains and Lambertian surfaces, and the approximate linearization is a poor approximation for non-black surfaces. Despite these limitations, the method's maturity makes it the ideal starting point for developing retrievals of 3D volumetric microphysical parameters at similar resolutions to those used in large-eddy simulations. The future spaceborne CloudCT mission (Schilling et al., 2019) will provide the required simultaneous multi-angle imagery for tomographic retrievals. Existing airborne instruments such as AirMSPI (Airborne Multi-angle Spectro Polarimetric Imager; Diner et al., 2013) and AirHARP (Airborne Hyper-Angular Rainbow Polarimeter; McBride et al., 2020) and the space-borne MISR (Multi-angle Imaging SpectroRadiometer) and MAIA (Multi-Angle Imager for Aerosols) also have the potential for tomographic retrievals, though they must additionally deal with the effects of cloud evolution (Ronen et al., 2021), as they do not acquire their observations simultaneously. The availability of these measurements makes the continued development of tomographic algorithms especially timely. Retrievals of this sort have the potential to provide the robust statistics of small-scale cloud and aerosol properties required for constraining cloud processes (Morrison et al., 2020), especially when extended to include information from other instruments such as cloud radar.

In this two-part series of papers, we present and validate an extension to the retrieval framework of Levis et al. (2020), which we have implemented and made publicly available in the software package Atmospheric Tomography with 3D Radiative Transfer (AT3D; Loveridge et al., 2022). This paper, which is Part 1, is devoted to the description of the retrieval methodology and the underlying theory of the retrieval, along with supporting numerical evidence. Part 2 of this study is devoted to tomographic retrievals on synthetic data to validate the method.

In Sect. 2, we describe the retrieval software AT3D, which is quite general in that it is designed for retrieving the 3D microphysical properties of external mixtures of atmospheric particles using multi-angle, multi-pixel, and possibly multi-spectral polarized radiances. In Sect. 3, we describe extensions to the method of Levis et al. (2020) to include an improved treatment of non-black surfaces and the retrieval of a plane-parallel medium in which the 3D domain is embedded, thereby improving the realism of the method. The Appendix documents the discrete implementation of the algorithm and model verification.

Despite the successful demonstrations of the tomographic retrieval (Levis et al., 2015, 2017; Martin and Hasekamp, 2018; Levis et al., 2020; Doicu et al., 2022a, b), it is still unclear how the effectiveness of tomographic techniques will vary with scattering regime. Previous studies have shown that success is not uniform, with poorer performance in optically thick clouds (Levis et al., 2015). It is not clear whether this is a result of a limitation in the approximate linearization method or a physical limitation. In Sect. 4, we present the theory of linear inverse transport problems and use it to explain the limitations of tomography in general, to provide insight into the physical processes that control the cloud tomography problem. This theory is drawn from both the wider literature on inverse problems (Bal and Jollivet, 2008; Bal, 2009; Chen et al., 2018; Zhao and Zhong, 2019) and the relevant literature in the atmospheric sciences that have studied the loss of information about spatial detail of cloud properties in multi-pixel radiances due to multiple scattering (Marshak et al., 1995, 1998b; Davis et al., 1997; Forster et al., 2020).

In Sect. 5, we perform a detailed quantitative validation of the approximate linearization to SHDOM that is utilized in AT3D, following Levis et al. (2020). Despite the success of the method in several test cases in Levis et al. (2015, 2017, 2020), no validation of the approximation itself has yet been performed, which has made it as yet unclear how the method will generalize to the wider variety of scattering regimes present in the cloudy atmosphere. We present an alternative derivation of the approximation that places it in the context of the forward-adjoint formalism developed by Martin et al. (2014). We can then explain the success of the approximate method using the theory presented in Sect. 4. In Sect. 6, we briefly quantitatively contrast stratiform and cumulus cloud geometries, in terms of the well-posedness of the tomography problem from a linear perspective, using the approximate Jacobian. We summarize our results in Sect. 7. In Part 2 of this study, we explore how these issues affect the fully nonlinear retrieval problem and demonstrate the effectiveness of the method described here.

Atmospheric Tomography with 3D Radiative Transfer (AT3D) is a software package designed to perform tomographic retrievals of atmospheric properties. It poses the inverse problem as a nonlinear, generalized least squares problem that is solved using iterative local optimization techniques. The solution procedure is physics based and uses the 3D RT model SHDOM (Evans, 1998) as its forward model to connect retrieved quantities to measured radiance. SHDOM is an explicit solver of the polarized 3D RTE that is well established in atmospheric science. During the intercomparison of 3D radiative transfer codes (I3RC; Cahalan et al., 2005), SHDOM was well within the consensus results. This is also true of intercomparisons of polarized RT (Emde et al., 2015, 2018).

In brief, SHDOM solves the integral form of the monochromatic vector RTE on a Cartesian grid using a fixed-point iteration scheme for collimated solar or thermal emission sources of radiation. A spherical harmonic expansion representation of the radiation field is used for computing the SOURCE function of the RTE, while a discrete ordinate representation is used for the streaming of radiation. At each iteration, the SOURCE function is transformed to discrete ordinates, new radiances are computed at each grid point using a short characteristic scheme, and a new spherical harmonic representation of the SOURCE function is computed. An adaptive spatial grid is employed so that grid cells with a variation in the SOURCE function larger than a threshold are split in half, generating new grid points. The number of spherical harmonics kept at each grid point is also adaptively truncated. SHDOM uses delta-M scaling (Wiscombe, 1977) and the truncated multiple-scattering (TMS) approximation (Nakajima and Tanaka, 1988) to treat problems involving highly anisotropic scattering. The truncation fraction used in the delta-M scaling of the optical properties is set by the angular resolution of the SHDOM solver and the Legendre expansion of the phase function.

AT3D builds on the software implemented by Levis et al. (2020), which itself builds upon the work of Evans (1998) in the publicly available Fortran implementation of SHDOM. AT3D is also a Python wrapper for the SHDOM RTE solver developed using the F2PY (Fortran to Python interface generator) tool (Peterson, 2009); this enables easy interfacing with external optimization libraries from SciPy (Virtanen et al., 2020). The use of Python also enables interactivity, even in high-performance-computing (HPC) environments, which accelerates data exploration and code prototyping. The key features of the SHDOM software are preserved in AT3D, with the only notable exception being that AT3D does not yet implement the message-passing interface (MPI)-based parallelization of SHDOM, so it is not yet able to efficiently utilize HPC resources to solve large-scale forward or inverse problems.

AT3D's strength is as a provider of a physics-based, and therefore flexible, method for solving the inverse problem of atmospheric tomography. It is therefore perfectly suited for performing sensitivity tests to changes in the measuring instrument's configuration (e.g., number of view angles and sensor resolution). AT3D supports the retrieval of multiple external mixtures of 3D distributions of scattering particles with solar and thermal sources, using arbitrary combinations of possibly polarized, multi-wavelength monochromatic radiances. Each unknown can be retrieved on a 3D grid or on a user-specified simplified spatial basis (e.g., column averages). AT3D does not yet support non-simultaneous measurements and the corresponding retrieval of a time-varying cloud (Ronen et al., 2021). Flexibility with the configuration of any retrieval problem is supported using object-oriented and functional programming in the Python wrapper. Currently, AT3D includes just the Rayleigh and Mie scattering particle models for homogeneous spheres distributed with SHDOM (https://nit.coloradolinux.com/shdom.html, last access: 20 March 2023; Evans, 1998). The parameterization of the size distribution and accompanying selection of unknowns (e.g., droplet number concentration or liquid water content) for retrievals is flexible.

We note that the software is far from a black-box tool and operates more as a library of high-level objects and functions that can be combined in short Python scripts according to user specifications. This level of flexibility is good for a research tool that is under active development, though it can lead to a steeper learning curve than in a well-defined executable with a fixed set of options common that is in other RT software packages. The code is well documented, and several tutorials are included with the code to mitigate this. Users or potential developers are welcome to contact the corresponding author to discuss their potential use case.

Just like the SHDOM software, AT3D is not a complete RT package and does not include detailed spectroscopic or particle scattering data that can be found elsewhere (Emde et al., 2016; Gordon et al., 2022; Saito et al., 2021). Interfacing with these packages is relatively simple, as the data are represented in AT3D using the xarray package (Hoyer and Hamman, 2017), which supports a variety of file formats such as NetCDF.

AT3D supports all surface bidirectional reflectance distribution functions (BRDFs) available in SHDOM but does not yet include linearization with respect to the parameters describing surface BRDFs. AT3D supports the inversion of optical or microphysical properties with solar, thermal, or combined sources. This is a helpful extension over Levis et al. (2020) for the far shortwave infrared (SWIR) and infrared (IR). However, linearization with respect to atmospheric temperature is not yet supported. These aspects are under development. The radiance calculations in AT3D have been generalized from SHDOM to support more realistic sensor geometries and sensor spatial response functions. However, as noted above, observables are currently monochromatic, and some small extension to the software would be required to accommodate observables requiring multiple monochromatic RTE solutions during inversion. The representation of phase functions within the SHDOM solver has been modified so that the SHDOM model is differentiable (Appendix B). The package also includes several useful tools, including a stochastic generator for making synthetic clouds (described in Part 2), a space-carving algorithm (Lee et al., 2018) for performing volume cloud masking for retrieval initialization (Sect. 5.3), and several basic regularization schemes.

While many practical extensions have been made to the retrieval software since Levis et al. (2020), including a comprehensive verification (Appendix C and E), which is critical to establish the veracity of the scientific results (Kanewala and Bieman, 2014), the novel elements of the retrieval software are the extensions of the linearization to non-homogeneous surfaces and open boundary conditions (BCs), which are documented in the following section.

We now present an overview of the iterative retrieval process. There are many mathematical terms introduced in this section. A glossary is provided in Appendix A for reference, and a flowchart of the retrieval process is shown in Fig. 1. The solution of the tomography problem involves the selection of a state vector (a) that parameterizes a discrete representation of the atmospheric optical or physical properties and best fits the available measurements (y) and any prior knowledge of the unknown state. The total size of the state vector (a) depends on the domain size and discretization scheme, but even for the smallest problems with a 3D gridded representation, it will typically range upwards of 10 000. We note that the state vector does not necessarily need to consist of physical variables and may instead consist of, for example, linear combinations of physical variables. This is a generalization that is preferable to ensure the associated optimization problem is well scaled (Nocedal and Wright, 2006).

The measurement vector (y) contains multi-angle, multi-pixel radiances, which may also be multi-spectral and polarized. The number of observations will also typically range above 10 000 and exceed the dimension of the state vector to avoid ill-posedness. The forward model F(a) provides the mapping from the state vector to the measurement space by producing synthetic measurements equivalent to y, based on the unknown state vector (a) and any fixed ancillary data. The forward model consists of multiple components, including the mapping from the state vector (a) to the 3D optical properties at all required wavelengths; the solution of the required 3D RT problems; and, finally, the sampling of the radiance field at the required positions, angles, wavelengths, and polarization states to produce synthetic measurements. We describe each of these components of the forward model in the following subsection.

We select a best-fitting state by minimizing a scalar cost function. The scalar cost function χ2 is chosen to penalize the misfit from the measurements in a generalized, least squares sense. 1χ2=y-FaTSϵ-1y-F(a)+Ra, where the error covariance matrix of the residual between the measurements (y) and the forward model F(a) is denoted by Sϵ and accounts for both measurement uncertainty and forward-model uncertainty. AT3D currently supports a block diagonal Sϵ, where error correlations are allowed between different Stokes components measured at each pixel, which supports the inclusion of certain types of forward-modeling error and instrumental noise (van Harten et al., 2018). It does not yet support systematic error correlations between pixels due to, for example, uncertainties in the flat-fielding operation, camera-to-camera intercalibration, band-to-band calibration error, or absolute calibration error. R(a) is a differentiable regularization term that reflects prior knowledge about the structure of the unknown state. Note that there is no requirement here that the regularization term takes the form of an a priori distribution. As such, the formulation in AT3D is more general than the optimal estimation frameworks utilized widely in atmospheric remote sensing (Sourdeval et al., 2013; Wang et al., 2016). We note that the cost function in Eq. (1) can take modified forms if transforms (e.g., logarithmic) are used to stabilize (pre-condition) the optimization process (Chance et al., 1997).

The solution to the inverse problem is given by the minimizer of the cost function, subject to the box constraints described by vectors of lower bounds (l) and upper bounds (u) on each element of the state vector. Formally, 2ã=argminaχ2,subject tol≤a≤u. For example, these bounds can be used to ensure positivity of the liquid water content or volume extinction coefficient. This optimization problem can be solved efficiently, using a local minimization technique, such as the limited-memory Broyden–Fletcher–Goldfarb–Shanno method for bounded minimization (L-BFGS-B; Byrd et al., 1995), when the cost function is differentiable. The local minimization proceeds through the selection of an initial guess a0 and iteratively updating the state through repeated evaluation of the cost function and its gradient. At the mth iteration, we then have the following: 3am+1=am+Δam. The retrieval method requires the selection of an initial guess, which can have a significant influence on the optimization. In Part 2 of this study, we describe some of the ways that initial guesses can be generated in AT3D.

The L-BFGS-B method is a quasi-Newton method which selects the update to the state vector (Δam) by first selecting a search direction through the minimization of an approximate, local, quadratic model of the cost function. The quadratic model uses an approximation to the Hessian of the cost function, which is formed by analyzing how the cost function gradient changes over the most recent M iterations. M is a hyperparameter of the optimization. In essence, the approximation to the Hessian of the cost function is formed from the finite differencing of successive gradient vectors. As such, it only reflects curvature information along the directions that the optimization trajectory is currently exploring. After the selection of a search direction, an inexact line search, obeying the Wolfe–Armijo conditions, which ensure stability and convergence, is then used to select the final update Δam along the search direction. The implementation of L-BFGS-B from the SciPy library is used (Virtanen et al., 2020). The L-BFGS-B algorithm has a better convergence rate than simple gradient descent but still only requires the evaluation of the cost function and its gradient. The computational cost of the update is modest once the gradient vector is computed. The storage requirement of L-BFGS-B is also modest and is limited to storing the state updates and gradient changes over the past M iterations, so it scales linearly with the size of the state vector. This makes the method appropriate for the large-scale optimization problem of cloud tomography.

The expression for the gradient of the data fit term required by the L-BFGS-B algorithm is as follows: 4∂χ2∂a=2y-F(a)TSϵ-1K, where K is the Jacobian matrix containing the partial derivatives of the ith output of the forward model with respect to the jth component of the state vector. This is determined as follows: 5Kij=∂Fi(a)∂aj. It is in our interest to understand the information content of our measurements and the factors that control the convergence rate of the retrieval so that these can be maximized. The linear information content in the measurements in the vicinity of a cost function minimum can be determined by using the Fisher information matrix, which, in the case of Gaussian errors, takes the simple form KTSϵ-1K. The singular-value spectrum of this matrix describes the magnitude of retrieval uncertainties across different directions in state space, and this distribution is largely controlled by the singular-value spectrum of K. Strong off-diagonal error covariances in the measurements can also affect the spectrum of the Fisher information matrix through Sϵ-1, though multi-angle imagers typically have block-diagonal error covariances that limit this effect. The largest physical values of the singular-value spectrum of K are physically bounded (Chen et al., 2018), and a linearization of RT tends to have rapidly (i.e., exponentially) decaying singular values (Culver et al., 2001). As such, we summarize the singular-value spectrum using just the condition number of the Jacobian matrix κ(K), which is the ratio of its largest (slargest) and smallest (ssmallest) singular values. 6κ(K)=slargestssmallest The L-BFGS-B method can suffer in systems where the condition number becomes large, where poor accuracy and slow convergence can occur (Zhu et al., 1997), even for quadratic cost functions. A slow convergence rate is particularly troubling, as it will determine the computational feasibility of performing cloud tomography. In Sect. 4, we relate the structure of the Jacobian matrix, specifically the condition number, to the properties of the medium through the principles of RT theory and support these principles with numerical simulations. We then examine the ability of the approximate Jacobian to accurately represent the true Jacobian matrix and to correctly capture the information content of the measurements about the state vector. Before presenting these results, we must first introduce the formulation of the forward model and the exact calculation of the Jacobian matrix.

Our analysis in this section is focused on describing the information content about the spatial variability in the optical properties contained within multi-angle measurements and therefore is focused on the inversion of the RT within a linearized context. We begin this analysis by first describing the conditioning of the exact Jacobian matrix, which we measure by its condition number (Eq. 6). Studying the exact Jacobian matrix provides a point of reference for understanding the effects of using approximate Jacobian matrix. Through the comparison, we can identify which limitations of tomography are physical and which are a result of using an approximate Jacobian. This section contains some results that have wider implications for tomographic retrievals and also some which are specific to the SHDOM solver. The implications of these results are discussed in Sect. 4.3.

The structure of the inverse problem and the nature of the Jacobian approximation used in AT3D can most easily be conceptualized if we use the equivalent adjoint formulation of the inner products, for reasons that we make clear below. This formulation will provide the basis for our presentation of the qualitative theory of inverse RT. Here we give only a brief summary of the adjoint formulation of the Jacobian calculations. More details on the formulation of the adjoint problem can be found in Martin et al. (2014) for 3D vector RT or in similar formulations for plane-parallel media (Hasekamp and Landgraf, 2005). In the adjoint formulation, the inner products in Eq. (51) are expressed in terms of an adjoint radiance field, whose sources are now the sensor response functions, pi and qi (instead of the Sun). The adjoint RT problem is solved using an adjoint solution operator U*, which is adjoint to the forward solution operator U. In practice, the adjoint RTE is not directly solved. Instead, a pseudo-forward RTE problem is solved, whose source is the direction-reversed and polarization-flipped sensor response functions pi† and qi†: 59pi†=Q†pir,-Ω, and 60qi†=Q†qir,-Ω, where 61Q†=111-1. The pseudo-forward problem is solved by the forward-solution operator U so it can, in principle, be solved by a forward model like SHDOM as long as its implementation supports sufficiently general volume and boundary source vectors. In practice, this pseudo-forward problem is much more numerically challenging due to the spatio-angular singularity of the sources pi† and qi† in 3D (Doicu and Efremenko, 2019; Martin and Hasekamp, 2018). The adjoint radiance field can then be found by direction reversal and polarization flipping of the pseudo-forward radiance I†r,Ω. 62I†r,Ω=Upi†qi† We can then express the inner products for the forward-model derivatives which form the elements of the Jacobian matrix as follows: 63∂Fi,n(a)∂aj=pi,∂Id∂ajDn×S2+qi,1,∂Id∂ajΓn+=Q†Ii†r,-Ω,ΔfjDn×S2+Q†Ii†r,-Ω,ΔgjΓn+. The spatial variations in the forward-model derivatives are controlled by the optical distance to the sensor, through Ii†, and also by optical distance to the solar source, through Δfj. Given the singular form of the sensor response functions, the RTE problem for the pseudo-forward radiance is a pencil beam illumination problem. This is a well-studied RT problem (Doicu et al., 2020; Liemert and Kienle, 2013; Martelli et al., 2016), as its solution is Green's function for 3D RT. The approximate Jacobian in the 3D RT domain is equivalent to approximating the pseudo-forward solution by its direct-beam solution. 64pi,TΔgj0r,ΩΓ1-r,Ω+SΔfjD1×S2D1×S2+qi,TΔgj0r,ΩΓ1-r,Ω+SΔfjD1×S2Γ1+=Q†T[pi†](r,-Ω),ΔfjD1×S2+Q†T[qi†](r,-Ω),Δgj0Γ1+ With the forward-adjoint formulation described above, each of the columns of the Jacobian matrix corresponds to a different observing geometry and therefore to a different pencil beam pseudo-forward RTE solution. We can use this to express the independence of columns in the Jacobian matrix in terms of the independence of the pseudo-forward-radiance fields, weighted by the radiance derivative source vectors Δfj and Δgj. In this way, we link the properties of the Jacobian matrix, such as its condition number, which controls the difficulty of the retrieval problem, to the optical properties of the medium using arguments from RT theory. To be clear, we use the linearized framework to classify the difficulty of the iterative optimization process based on the current state vector. The linearization around the ground truth does provide some information on the difficulty of the inverse problem, but we must bear in mind that, given the non-convex nature of the inversion, the iterative retrieval may encounter a difficult cost function structure far from the ground truth, depending on the optimization trajectory. This linearized framework provides us with some hypotheses about the behavior of AT3D's tomographic retrieval that can be used to explain the behavior of fully nonlinear retrievals in terms of physical principles.

With the theory introduced in the previous section in place, we can examine the consequences of the approximation to the Jacobian (Eq. 58) used in AT3D and how that approximation exhibits itself as quantitative errors. We examine how well the approximate Jacobian reproduces the behavior described in Sect. 4. As described in Sect. 3.3, we are approximating the pseudo-forward radiance by its direct beam (Eq. 61). In this case, we can see that the entire diffuse pseudo-forward-radiance profile (Fig. 7) will be neglected. This means that the approximate Jacobian does not represent the non-local sensitivity of measurements to changes in optical properties outside of their field of view, other than through changes to the direct solar transmission. This may seem a rather extreme approximation, but we must bear in mind that the most stable information to extract is contained in the highly localized direct-beam component (Bal and Jollivet, 2008). The success of the approximation requires that the pixel-to-pixel smoothing only becomes important and significant when the medium is optically thick enough so that there will be significant decay in the sensitivity with distance from the sensor. In this case, the loss of sensitivity to the cloud interior from all pixels (in the linearized setting) is much more important than neglecting the pixel-to-pixel smoothing. In this section, we test the extent to which this is true quantitatively.

From the theory developed in the previous section, we also expect a sensitivity of the approximate Jacobian to the dimensionality of the RT problem. In 3D, the pseudo-forward-radiance field is highly singular in space (Fig. 7) and anisotropic. On the other hand, as the dimensionality of the transport decreases, the pseudo-forward solution will become increasingly less singular. For example, in 1D, the pseudo-forward source is a plane illumination, just like the Sun (Hasekamp and Landgraf, 2005). The symmetry of the source means that the pseudo-forward radiance does not disperse perpendicular to the collimated source, but instead, it substantially modifies the depth dependence of the pseudo-forward-radiance field and reduces its anisotropy. This means that the direct-beam approximation will perform worse, and the performance of the approximation to the Jacobian calculation adopted here cannot be expected to generalize from 3D to 2D or, especially, 1D problems. This has an important implication for our extension of the approximate Jacobian to the linearization of the system of RTEs described in Sect. 3 to model a 3D domain embedded in a plane-parallel atmosphere. As such, we separate our error analysis between the elements of the approximate Jacobian corresponding to the primary 3D domain D1 (Fig. 2) and the other elements controlling the boundary radiances through the auxiliary RTE problems.

Note that, in the following analysis, we quantitatively validate derivatives of radiances only with respect to volume extinction coefficient at different grid points rather than other optical properties such as the single-scattering albedo or components of the phase matrix. The approximation to the Jacobian only approximates the pseudo-forward radiance and therefore is common to derivatives of radiances with respect to all optical properties. However, each optical property interacts differently with the pseudo-forward-radiance field. For example, the single-scattering albedo interacts with all spherical harmonics of the radiance field, while the extinction coefficient interacts with all except the isotropic component. As such, errors in radiance derivatives with respect to single-scattering albedo may be disproportionately affected by error in the pseudo-forward radiance that occur optically far from the sensors compared to derivatives with respect to extinction. In contrast, the higher-order expansion coefficients of the phase matrix are only sensitive to the high-order spherical harmonics of the radiance field and will therefore be disproportionately affected by errors in the pseudo-forward radiance that occur optically close to sources. As a result, Jacobian errors may be distributed slightly differently in space and observation angle as the angular structure of Δfj changes with the nature of the optical (or microphysical) unknown. We focus solely on extinction to illustrate how errors in the approximate pseudo-forward radiance propagate to the Jacobian as the retrieval of extinction is fundamental for retrievals of spatial structure.

To quantify the agreement between the finite-differenced Jacobians defined in Sect. 4.1 and the equivalent approximate Jacobians (K̃), we use the relative Frobenius error, which is an element-wise relative root mean square error (RMSE): 67relative Frobenius error=K̃-KFKF=∑ijK̃ij-Kij2∑ijKij2. Our validation approach of comparing the approximate Jacobian to the finite-differenced Jacobians measures the consistency of the forward model with the approximate Jacobian. The consistency between these two is what is important for the robustness of the local optimization method used in the tomographic retrieval implemented in AT3D. As such, the relative Frobenius error includes the effects of computational noise in the forward model that will emerge in the finite-differenced derivatives. When evaluating derivatives calculated using a forward-adjoint principle, the relative Frobenius error was around 0.04, reflecting uncertainties in the finite differencing and inconsistencies between the forward and adjoint models (Doicu et al., 2020). We expect a similar degree of accuracy in the optically thin limit for the approximate Jacobian. We performed extensive verification of the approximate Jacobian. We identified a number of instances where the computational noise in the finite-differenced derivatives is actually the dominant source of error, based on comparison against an analytic solution (Appendix E). For that reason, it is important to keep in mind that this error metric is a measure of the consistency of the approximate Jacobian with the forward model and that the stability of the forward model itself is also contributing.

We can make use of the computational efficiency of the approximate Jacobian to explore the dependence of the condition number of the Jacobian on the spatial structure and optical thickness of the cloud field in more detail than in Figs. 5 and 13. In particular, we illustrate the critical importance of the spatial structure of the extinction field for determining the feasibility of tomography. We contrast the behavior of plane-parallel homogeneous clouds and 3D Gaussian extinction fields, which are restricted to a (1 km)3 domain under inversion. Note that the plane-parallel homogeneous cloud is still resolved in 3D, and the radiation transport is in 3D. We are examining our ability to retrieve inhomogeneities embedded within the cloud. As such, the dimensionality effects described in Sect. 5 are not at play here. We use the same hemispheric set of observations and the same grid geometry used in Sects. 4.1 and 5.1. The clouds are conservatively scattering and use the Mie phase function with a black surface. We calculate the derivatives with respect to all internal grid points in each horizontal grid point but only every second grid point in the vertical. This is done to avoid the need to adopt special techniques for the singular-value decomposition (SVD) of large matrices to calculate the condition number.

We classify the plane-parallel clouds by their vertical optical depth and the 3D Gaussian extinction fields by their vertical optical depth to the center of the cloud (half of their diameter). We refer to this as the “optical dimension”. This classification puts the two types of extinction field on the same footing in terms of the minimum transmission from any sensor to the grid point that is optically furthest from all sensors. They are therefore equivalent in terms of the presence of a veiled core, as defined by Forster et al. (2020) and investigated in Sect. 4.2.