A comprehensive inter-comparison of seven radiative transfer models in the limb scattering geometry has been performed.
Every model is capable of accounting for polarization within a spherical atmosphere.
Three models (GSLS, SASKTRAN-HR, and SCIATRAN) are deterministic, and four models (MYSTIC, SASKTRAN-MC, Siro, and SMART-G) are statistical using the Monte Carlo technique.
A wide variety of test cases encompassing different atmospheric conditions, solar geometries, wavelengths, tangent altitudes, and Lambertian surface reflectances have been defined and executed for every model.
For the majority of conditions it was found that the models agree to better than 0.2

The limb scattering measurement technique involves viewing through the side, the limb, of the atmosphere while measuring scattered sunlight (see Fig.

The limb-viewing geometry and definitions of solar zenith angle, solar azimuth angle, solar scattering angle, and tangent altitude. The tangent altitude and solar zenith angles are defined relative to the un-refracted tangent point such that a solar scattering angle of 0

Several satellite-based limb scattering instruments have flown in the past few decades.
Notably, the Optical Spectrograph and InfraRed Imager System

Vertical profiles of limb scattering spectra can be inverted to obtain distributions of atmospheric constituents with spectral absorption or scattering features.
These include but are not limited to stratospheric aerosol

All of the aforementioned retrieval methods use RTMs operating in scalar mode, where only the intensity,

Several studies have been performed which inter-compare the accuracy of polarized RTMs

The most comprehensive previous inter-comparison focusing on the limb scattering geometry was performed by

This study serves to both update the state of inter-comparison of RTMs in limb scattering geometry and to improve on it in several ways.
Firstly, all participating RTMs simulate polarization in the atmosphere and provide full Stokes vectors which are compared; these results are of significant importance for the upcoming ALTIUS mission.
Secondly, the treatment of stratospheric aerosols is updated to use Mie scattering solution rather than a Henyey–Greenstein phase function; the Mie scattering treatment is more representative of the current state of limb retrievals

Descriptions of each model are presented in Sect.

Generally, modern RTMs include a variety of tools to aid in specifying the atmospheric state and the viewing geometry.
These could be relatively simple things such as pre-computed climatologies of pressure, temperature, and ozone, or something more involved such as a built-in Mie scattering code to calculate the optical properties of stratospheric aerosol particles of a given size distribution.
However, the core purpose of every RTM is to solve the radiative transfer equation.
Some models may contain several algorithms to do this, and each one is called a

RTMs typically belong to one of two classes: statistical or deterministic. Statistical models solve the radiative transfer equation using Monte Carlo simulation of photon paths through the atmosphere, while deterministic models use discretization, interpolation, and various simplifying assumptions. Statistical models are often easier to implement since fewer assumptions are made; however they usually are orders of magnitude slower computationally.

Deterministic models solve the radiative transfer equation (RTE) using some form of numerical integration over the line of sight (LOS) and by making various simplifying assumptions. The choice of how and which quantities are discretized can result in completely different methods being applied to solving the RTE.

These methods can further be classified according to how the sphericity of the atmosphere is handled when calculating the multiple-scattered radiance field. Plane-parallel models assume a flat Earth and can therefore not be applied to simulate the limb-viewing geometry. Pseudo-spherical models employ a plane-parallel solution but initialize the data in the RTE with the solar irradiance attenuated through a spherical atmosphere. Approximate spherical models trace the observer LOS through a spherical atmosphere, calculate the single-scatter term spherically, and then use an approximately spherical multiple-scatter source function for light that has been scattered more than once (typically from one or more pseudo-spherical calculations, although the exact method may vary from model to model). Lastly, fully spherical models account for sphericity in all aspects of the calculation.

Some of these approximations have been shown to have significant systematic effects on calculated radiances in the limb-viewing geometry.
Most notably, approximate spherical methods which use a single plane-parallel solution for the multiple-scatter source have been shown to be systematically high (on the order of 5

The Gauss–Seidel limb scattering (GSLS) RTM builds upon the techniques described by

GSLS has been used in several projects involving the retrieval of atmospheric constituents from limb scatter measurements.
Most notably, GSLS is currently used as the RTM for the operational version of the OMPS-LP ozone and stratospheric aerosol data products

SASKTRAN is a fully spherical, vectorial RTM originally developed at the University of Saskatchewan to process data from the Optical Spectrograph and InfraRed Imaging System

SASKTRAN-HR has various options that control the accuracy of the solution, but the main one is the number of diffuse profiles, i.e., the number of discretizations used in solar zenith angle to compute the multiple-scatter field.
The model has been configured to use the number of diffuse profiles required to obtain approximately 0.2

The SCIATRAN software package provides tools for modeling radiative transfer processes in the ultraviolet to thermal infrared spectral range.
A detailed review of available algorithms, selected comparisons, and applications is given by

In the limb-viewing geometry SCIATRAN can operate in two modes, fully spherical and approximately spherical. In the approximately spherical mode, the single-scatter radiance is calculated accounting for the full sphericity of the Earth, while the multiple-scattered signal is approximated by several pseudo-spherical calculations. In fully spherical mode, the approximately spherical solution is iterated in a fully spherical geometry to account for sphericity effects. SCIATRAN uses the fully spherical mode in the shown comparisons.

SCIATRAN accounts for refractive effects and calculates approximate weighting functions.
SCIATRAN has been used in numerous applications spanning multiple research areas.
One of SCIATRAN's primary applications is its use as the forward model for SCIAMACHY limb scatter retrievals of including, but not limited to, ozone

Statistical models use Monte Carlo (MC) simulation to solve the RTE. In spherical geometry, the most common technique is the so-called backward MC method, or adjoint method. Here, photons are traced, starting at the sensor, through the atmosphere and towards the sun; this is in contrast to the forward method, where photons originate at the source (Sun). Along the photon path, the choice of where the photon scatters and the direction of scattering are sampled based on the probability of a scatter event occurring. The final radiance, and associated precision, are estimated by analyzing an ensemble of a large number of photons. All of the models within this study use a variant of the backward MC method.

The MC technique naturally has few assumptions, which allows for easier implementation of new features. A primary example of this is implementing atmospheric constituents that vary in three dimensions, rather than only in altitude. It is also quite natural to handle the full sphericity of the atmosphere. Because of these reasons, statistical models are primarily used as benchmark models and to study new effects. Statistical models are typically orders of magnitude slower than deterministic models and are thus usually not used in operational retrieval methods. They also contain inherent random noise, driven by the number of photons used in the simulation, which may need to be considered depending on the application.

The Monte carlo code for the phYSically correct Tracing of photons In Cloudy atmospheres (MYSTIC) model

The SASKTRAN RTM contains a MC mode (SASKTRAN-MC) based upon the Siro algorithm

Siro is a statistical, fully spherical, polarized RTM developed at the Finnish Meteorological Institute, using the backward MC method

SMART-G (Speed-up Monte-carlo Advanced Radiative Transfer code with GPU) is a radiative transfer solver for the coupled ocean–atmosphere system with a wavy interface

The radiances at any level of the domain can be estimated using the local estimate variance reduction method

There is a subtle difference in the way the backwards Monte Carlo method is implemented that can be noticed in the subsequent comparisons. One option is to trace rays through the atmosphere and calculate the scattering probability at each layer interface. This gives the possibility of photons not scattering and directly escaping the atmosphere, which is important for estimates of radiative fluxes (not directly applicable for limb scatter measurements). A consequence of this is that at longer wavelengths and higher tangent altitudes where the atmosphere is optically thin, a large number of photons are required to reduce the statistical noise to acceptable levels. In this study this technique is used by MYSTIC and SMART-G.

An alternative technique is to force every photon traced backwards from the observer to scatter. Random numbers are generated to determine the scatter location, not if scattering actually occurs. Photons can then be weighted by the optical thickness to account for the probability of scattering. A benefit of this technique is that the number of photons required to hit a desired noise floor is more uniform in wavelength and altitude space, but the technique is more specific to limb scattering measurements. Siro and SASKTRAN-MC both use the same technique where every photon traced is forced to scatter.

SMART-G includes an option to force additional scattering in limb mode, but only for the first (single) scatter. The option has a similar effect to the technique used by Siro/SASKTRAN-MC in that it reduces the variance of the calculation for optically thin scenarios, but it is not exactly equivalent.

Overall, all of the mentioned techniques solve the radiative transfer equation with the same level of accuracy.
The only difference is the number of photons required to reach a desired level of precision.
A more in-depth discussion of the computational efficiency of the different techniques for different scenarios is presented in Sect.

Test cases are designed to explore the aspects of the RTMs that are applicable for past, present, and future satellite-based limb-scattering measurements.
All tests are performed for the following range of tangent altitudes, solar angles, surface reflectance, atmospheric constituent conditions, and wavelengths:

80 tangent altitudes from 0.5 to 79.5

9 combinations of SZA and SAA which are given in Table

3 values of a Lambertian effective reflectance of 0, 0.3, and 1.

3 atmospheric constituent conditions: pure Rayleigh scattering, Rayleigh scattering and ozone absorption, and Rayleigh

11 wavelengths provided in Table

Solar zenith angles, solar azimuth angles, and solar scattering angles used in the test cases.

The Rayleigh scattering cross section, ozone absorption cross section, and stratospheric aerosol refractive index used for the test cases.

In addition to different atmospheric and geometry conditions, test cases are selected using different RTM settings:

single scattering only, vectorial, no refraction;

multiple scattering included, scalar, no refraction;

multiple scattering included, vectorial, refraction.

One of the challenges in performing an inter-comparison of RTMs is ensuring that the inputs are the same across every model. In this study, care was taken so that the input parameters were specified in a way that can be assimilated by every model in the study.
Stratospheric aerosol is specified as a log-normal distribution of Mie scattering particles with a median radius of 80

The background atmosphere is specified on a 1

As noted in previous polarized RTM intercomparisons

Sign that each model's Stokes parameters were multiplied by to follow the definition of

The test cases specify the atmospheric state parameters on a 1

The RTMs GSLS, SASKTRAN (HR and MC), and Siro assume linear interpolation and handle it through analytic methods.
All four models calculate optical depth exactly, assuming a linearly varying extinction

The other RTMs (MYSTIC, SCIATRAN, SMART-G) use homogeneous layers.
To better harmonize the treatment of the input data, the RTMs using homogeneous layers have been configured using sub-layers with linear interpolation.
Figure

The main challenge in interpreting and attributing differences between models is that the true answer is not known.
All comparisons shown in this section are relative to what we have called the multi-model mean (MMM).
The MMM is composed of the set of models that agree with each other to a level that cannot be attributed to a concrete difference in a single model.
For the single-scattering test cases the level was determined to be 0.2

While tests were performed for both vectorial and scalar modes, no significant differences were found in agreement between the models in scalar or vectorial mode. Therefore, for the purpose of brevity, no comparisons of strictly scalar mode are shown. It is understood that the results for the polarized comparisons are equally applicable to scalar calculations.

Simplifying assumptions made for the single-scatter calculation are minimal, and thus we expect differences to be relatively small.
We do not expect zero differences due to the various methods of gridding in the vertical dimension of the atmosphere.
As originally done in the limb geometry by Siro

Differences for the most extreme single-scatter case (Rayleigh

Percent differences in single scatter

The excellent agreement in single scatter is expected due to the relative simplicity of the calculation.
Fundamentally each RTM solves the single-scatter problem in the same way with minimal assumptions. The primary purpose of this test is to ensure that the inputs to RTM are configured correctly.
The agreement of 0.1

Differences in radiance with multiple scattering enabled are expected to be larger than those seen in single scatter owing to the extra complexity of the radiative transfer problem. The discrete models must deal with discretizations of the multiple-scattering source term and may also make fundamental approximations for the sake of computational speed. Comparatively the statistical models employ a simpler technique.

In Fig.

Percent differences in

GSLS at 351

SASKTRAN (HR and MC), SCIATRAN, SMART-G, and MYSTIC all agree for all conditions to better than 1

While all comparisons so far have involved vector radiative transfer calculations, only differences in the

Leftmost column: the MMM calculation of

Generally agreement between

In order to isolate the polarization effects, we consider two quantities: the degree of linear polarization (DOLP) and the linear polarization orientation (LPO).
DOLP is defined as

Absolute differences in DOLP relative to the MMM (MYSTIC, SASKTRAN-MC, and SMART-G) for each RTM at a variety of wavelengths and solar conditions. The atmospheric optical properties include Rayleigh scattering, ozone absorption, and stratospheric aerosol Mie scattering. Refraction is disabled. Dashed vertical lines indicate

Absolute differences in LPO [

Differences in the DOLP and LPO are overall small, with a few exceptions.
Once again, the MC models MYSTIC, SASKTRAN-MC, and SMART-G agree in all cases to the level of statistical noise in the computation.
At 300

Siro has differences in DOLP but curiously does not have any significant differences in LPO.
At 351

Both SASKTRAN (HR and MC) and GSLS make the assumption that

All of the models considered thus far with the exceptions of SASKTRAN-MC and MYSTIC support atmospheric refraction to some level. While Siro has support for refraction, it was not tested as part of this study. GSLS and SASKTRAN-HR neglect refraction of the incoming solar rays. Furthermore GSLS and SASKTRAN-HR neglect refraction for multiple-scattering effects, only implementing refraction for the line-of-sight ray. SCIATRAN and SMART-G implement refraction in a generic way accounting for all solar and multiple-scattering effects. SASKTRAN-HR has since been updated to include refractive effects for incoming solar rays and multiple-scatter effects, but the calculations here use only line-of-sight refraction.

Differences in radiance when refraction is enabled for the stratospheric aerosol scattering case are shown in Fig.

Same as Fig.

To further investigate these differences, the ratio of refracted to unrefracted

Ratio of refracted to unrefracted

There are several possible reasons for the small observed differences between GSLS, SASKTRAN-HR, and SCIATRAN.
The index of refraction of the atmosphere was not harmonized between the models; instead each model performed internal calculations using the provided atmospheric temperature and pressure.
Various methods exist to do this calculation, and they may not be the same between each RTM.
Differing methods of ray tracing and integration can also lead to small differences.
Since SCIATRAN (and SMART-G) included refractive effects for the incoming solar ray and multiple-scatter terms, it is possible that solar refraction could be the source of some minor differences.
The solar geometries tested here have been limited to SZA

We have considered a basic run time comparison between the models in the study. While a useful exercise, there are technical challenges in standardizing the hardware used to execute the models, and in the case of SMART-G, which uses a GPU, the standardization is not possible. More importantly every model has settings that involve an accuracy–speed trade-off. For example, with the deterministic models, there are various discretization settings that can dramatically affect the speed of the calculations.

Harmonizing the balance between accuracy and speed between all of the RTMs is impractical; instead we opt for a simple order of magnitude timing estimate. The time taken to execute all of the multiple-scatter-enabled, polarized tests for each model without refraction is shown in Table

Estimated time to execute all multiple-scattering-enabled, no-refraction tests. This includes 3 effective surface albedos, 3 different atmospheric compositions, 11 wavelengths, 9 solar geometries, and 80 lines of sight. The MC models were executed to the precision shown in Fig.

Analyzing the timing of the Monte Carlo models is inherently more challenging as the calculations also contain statistical noise.
It is common to benchmark models for a set number of photons; however the number of photons used is not comparable between Siro/SASKTRAN-MC and MYSTIC/SMART-G since the MC technique is not the same.
Instead, the precision of the calculation must be directly compared, which is shown for a typical condition in Fig.

Precision estimates for the MC models with multiple scattering, Rayleigh scattering

For the given precisions, the CPU-based statistical models, MYSTIC, SASKTRAN-MC, and Siro, have run times within an order of magnitude.
The run time of Siro appears large; however the precision is generally better. Approximately scaling the Siro calculation to 0.2

The run time for the GPU-based MC model, SMART-G, is

A systematic comparison has been performed between seven radiative transfer models operating in the limb scatter geometry. The seven models are capable of handling the sphericity of the atmosphere and compute the Stokes vector accounting for polarization. The test cases cover a wide variety of solar angles, Rayleigh scattering, ozone absorption, Mie scattering, and surface reflectances.

In single scatter, the deterministic models GSLS, SASKTRAN-HR, and SCIATRAN agree within 0.1

For almost all conditions with multiple scattering enabled, the agreement between the fully spherical models is within 1

Siro can have disagreement of up to 3

At longer wavelengths in all atmospheric conditions and when the Lambertian surface reflectance is high, GSLS shows biases of up to 3

Refraction has been tested for GSLS, SASKTRAN-HR, SMART-G, and SCIATRAN.
The refractive effect among all models is almost indistinguishable, with the exception of a

Differences in quantities representing linear polarization, the DOLP and LPO, have also been assessed; however the results are more difficult to interpret.
The MC models MYSTIC, SASKTRAN-MC, and SMART-G agree within statistical noise for all considered conditions and serve as a combined reference.
SASKTRAN-HR and SCIATRAN generally agree with the reference at a level of 0.002 in DOLP and 0.2

Overall the agreement between the models is excellent and is better than has been reported for scalar comparisons in the past. The agreement provides additional confidence in the retrievals from limb scatter instruments such as OMPS-LP, OSIRIS, and SCIAMACHY. In particular, confidence in modeling the polarized signal is important for the upcoming ALTIUS mission.

There are several areas where future studies comparing RTMs in the limb-viewing geometry could expand upon. Scattering from larger, non-spherical, particles, droplets, or crystals such as those contained in clouds should be assessed, which may result in larger differences in particular for circular polarization. More extreme cases with higher solar zenith angles may be checked to further push the models, which could also be used to determine the effect of refraction at higher solar zenith angles. Non-Lambertian reflecting surfaces as well as a larger variety of stratospheric aerosol conditions would be another interesting area of study. Finally, the impact of the observed differences between the models could be studied in the context of standard applications such as limb scatter species retrievals.

The input atmospheric data, test cases, and the results for each model are made publicly available as

DZ wrote the initial draft of the manuscript. Model runs were performed by DZ, GF, RL, AM, and AR. All authors contributed in designing the study, interpreting the results, and revising the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “New developments in atmospheric limb measurements: instruments, methods, and science applications (AMT/ACP inter-journal SI)”. It is a result of the 10th international limb workshop, Greifswald, Germany, 4–7 June 2019.

Robert Loughman appreciates the support of Ghassan Taha and Tong Zhu for the code testing work done for this project. Dan Kahn, Jason Li, Mike Linda, and Colin Seftor also provided valuable assistance with NASA computer access, code setup, and timing assessments, while Surendra Bhatta contributed Python programming assistance. The manuscript was greatly improved by the helpful suggestions of the two reviewers (Christopher Sioris and Chris McLinden) as well as those from Sergey Korkin.

The work has been partially supported by the Canadian Space Agency, the European Space Agency, the state of Bremen and the University of Bremen. Advancements of the radiative transfer model SCIATRAN made are a contribution to the project VolARC funded by the German Research Foundation (DFG) through the research unit VolImpact (grant no. FOR2820). The work by Antti Mikkonen was supported by the Academy of Finland Centre of Excellence in Inverse Modelling and Imaging (project number 312125). The work of Robert Loughman was funded by NASA contract (grant no. 80NSSC18K0847; led by Ghassan Taha).

This paper was edited by Chris McLinden and reviewed by Chris Sioris and Chris McLinden.