The Optical Spectrograph and InfraRed Imaging
System (OSIRIS) instrument on board the Odin spacecraft has been
measuring limb-scattered radiance since 2001. The vertical radiance
profiles measured as the instrument nods are inverted, with the aid
of the SASKTRAN radiative transfer model, to obtain vertical
profiles of trace atmospheric constituents. Here we describe two
newly developed modes of the SASKTRAN radiative transfer model:
a high-spatial-resolution mode and a Monte Carlo mode. The
high-spatial-resolution mode is a successive-orders model capable of
modelling the multiply scattered radiance when the atmosphere is not
spherically symmetric; the Monte Carlo mode is intended for use as
a highly accurate reference model. It is shown that the two models
agree in a wide variety of solar conditions to within 0.2 %.
As an example case for both models, Odin–OSIRIS scans were simulated
with the Monte Carlo model and retrieved using the high-resolution
model. A systematic bias of up to 4 % in retrieved ozone number
density between scans where the instrument is scanning up or
scanning down was identified.
The bias is largest when the sun is near the horizon and the solar
scattering angle is far from 90

Remote sensing has played an integral role in our understanding and
monitoring of Earth's atmosphere, notably in the study of ozone and
the retrieval of vertically resolved atmospheric constituent profiles.
Some of the first standard ozone profiles were retrieved using data
from occultation instruments which provided high-quality, near-direct
measurement of optical depth profiles. Although highly accurate,
these instruments had limited sampling capabilities, generally
measuring between 16 and 32 profiles per day. To help address this,
several instruments that measure limb-scattered light in the
ultraviolet (UV) to near infrared (NIR) have since been placed in
orbit, including SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY (SCIAMACHY)

While limb-scatter measurement provides greatly improved sampling
rates, the signal interpretation is much more convoluted than for
occultation measurements, owing to the complicated scattering paths of
UV and visible light. Nevertheless, several successful retrieval
algorithms have been implemented by the SCIAMACHY

This paper describes the addition of two new engines to the SASKTRAN framework which allow for Monte Carlo and high-spatial-resolution radiative transfer modelling. As an example of usage, systematic errors in the OSIRIS ozone retrieval due to low-resolution radiative transfer limitations are explored and results from model simulations are used to identify and improve treatment of problematic measurement conditions.

The forward model used in this study is SASKTRAN. SASKTRAN is a radiative transfer framework consisting of two major components: a set of climatologies and optical properties which are used to specify the atmospheric state and an engine which solves the equation of radiative transfer for quantities of interest. Currently SASKTRAN consists of three separate engines: a standard successive-orders-of-scattering engine (SO), a high-spatial-resolution engine (HR), and a Monte Carlo engine (MC). All components of the SASKTRAN framework treat the planet and atmosphere as spherical, and all path lengths and angles are computed using a spherical geometry.

The original SASKTRAN radiative transfer model outlined in

The limb-scatter geometry used in SO, HR, and MC. The solar viewing angles are defined at the tangent point.

The radiance can be written in integral form,

The radiance is calculated with the successive-orders-of-scattering
method. Ignoring the discretization that needs to be done in a real
model, the technique has an intuitive explanation beginning with the incoming solar irradiance.
Solar rays are attenuated to all points in the atmosphere and
scattered, forming the source function for light that has been
scattered once. The scattered rays are then propagated through the
atmosphere and, once again, scattered at all points, forming the
source function for light that has been scattered twice. The process
can then be repeated to find the source function for light that has
been scattered to an arbitrary order. Mathematically we are applying
Eqs. (

Once the full source function is known, the radiance for a specific line of sight
can be calculated through a relatively simple line integral. We approximate
the line integral by splitting the ray into segments where the extinction
and source function are assumed to be constant. We call these segments

To perform the actual calculation of the source term, the integral in
Eq. (

Previously, SO would approximate the diffuse field by assuming the
diffuse profiles are uncoupled. For example, when calculating the
third-order-of-scatter source term a diffuse profile uses only its own
second-order source term, rather than coupling to other profiles. The
approximation affects the third and higher orders of scatter, and is
thus small at many wavelengths, but can be significant in certain
conditions, for example, near 350

A new high-spatial-resolution engine under the SASKTRAN radiative transfer framework has been developed. The engine is intended for use in future satellite missions requiring higher detail in the radiative transfer calculation. The radiative transfer equation is solved in the same fashion as SO, but less information is cached for each wavelength. The reduction in caching causes HR to use approximately one-seventh the RAM in identical configurations, at the expense of increased execution time.

Lower memory usage allows for higher-accuracy computations in both the
single-scattered and diffuse radiation fields. In addition, several
new features have been implemented:

the ability to handle areas of large or highly variable
extinction (e.g. cirrus clouds, see

support for atmospheric constituents which vary in two or three dimensions, e.g. latitude and longitude, rather than exclusively in altitude;

weighting functions for absorbing species can be approximated analytically in one and two dimensions for little computational cost.

Line integrals must be performed in two different areas when
performing the successive-orders method: the calculation of optical
depth and the integration of source terms along a path. Optical depth
is calculated as in

When the extinction varies significantly between

The radiance along a specific line of sight as a result of atmospheric
scattering may be written as the sum of radiance contributions from
individual cells attenuated back to the observer,

Support has been added in HR mode for the atmospheric constituents to vary in two or three dimensions. There are two main complications in breaking the assumption of horizontal homogeneity. First, the diffuse field now varies in an additional dimension; second, the line integration techniques need to be modified to deal with an additional dimension in which quantities may vary.

To account for the now five-dimensional diffuse field, diffuse profiles are not limited to placement at discrete solar zenith angles. Interpolation of the source function between diffuse profiles is done by finding the nearest three diffuse profiles and performing linear interpolation using the vertices of the formed triangle.

For a limb geometry measurement, simply finding intersections with
a set of spherical shells, as is done in SO, causes cells near the
tangent point to have lengths of up to

The two- and three-dimensional atmospheric grids used in HR.

There are three primary modes where the HR model supports variation of
atmospheric constituents in more than one dimension. The first is the
fully three-dimensional mode, wherein atmosphere is allowed to vary
arbitrarily. Internally, the atmosphere is stored as a set of
vertical profiles, specified above discrete geographic locations. For
HR it is sufficient (and desirable, for time efficiency) to specify
the atmosphere only on a region slightly larger than that where the
diffuse field is to be solved. The Delaunay triangulation on a sphere
of atmospheric profile locations is found, and queries of the
atmospheric state are answered by interpolating between the three
profiles which, when their locations are joined by geodesics to form
a spherical triangle, bound the query point

For satellite tomography applications, a second mode is implemented
where the atmosphere varies in the orbital plane, i.e. in altitude and
in angle along the orbit track (Fig.

As previously stated, the assumption of horizontal atmospheric homogeneity leads to the simplification that the diffuse field does not vary in solar azimuth. This simplification also holds when atmospheric constituents are allowed to vary in solar zenith angle as well as altitude. This special case is particularly useful for the inclusion of photochemically active species. Here, diffuse profiles can be placed once again in solar zenith angle without compromising the accuracy of the solution. To account for the additional variation in the numerical integration, cones of constant solar zenith angle are added to the ray tracing primitives list.

The HR model adds the capability to calculate weighting functions
(derivatives of radiance with respect to atmospheric parameters)
analytically with little computational overhead. Fast calculation of
weighting functions is necessary for many retrieval algorithms. One
method to compute the weighting functions is through finite-difference
schemes, which requires the forward model to be run a second time with
an atmospheric parameter slightly perturbed. Often when calculating
weighting functions the forward model is run for single-scattering
only to save on execution time. The single-scattering approximation
was shown to produce weighting functions sufficient for use in

Here we present a simple method for analytical computation of
weighting functions which is fast, is more accurate than the
single-scattering approximation, and extends naturally to two- and
three-dimensional atmospheres. We start by taking the derivative of
Eq. (

For absorbing species, i.e.

Ozone weighting functions at a wavelength of
330

As an example, weighting functions for a typical ozone distribution
were calculated for a line of sight with tangent altitude of
24.5

For a single-scattering species,

As shown in

It is desired, therefore, to test the discrete-ordinates
successive-orders method as implemented in SO and HR while preserving
the underlying framework of atmospheric state, optical properties,
climatological species, ray tracing, and numerical integration. This
motivates the development of the Monte Carlo engine, which uses
optical properties, ray tracing algorithms, and quadrature identical
to that of SO and HR (including those developments noted in
Sects.

The backwards Monte Carlo algorithm for observers with a narrow field
of view, as implemented in several radiative transfer codes

The exactly

Similarly, for scalar light (recall the scalar phase function depends
on scattering angle only) the

Estimates of

Higher-order radiance

Following the backwards Monte Carlo algorithm, the ray history begins
at the observer, with transmission along the observer line of sight
providing the distribution

The algorithm is multithreaded over ray histories. That is, each
thread propagates a separate ray history to

Because MC resolves rays at every scattering event, it is simple to
collect statistics about the physical distribution of scattering
points as well as the variance and covariance of different orders of

All timing is carried out on an Intel Core i7-4770 CPU at

Timing of the Monte Carlo engine is highly sensitive to wavelength and
solar zenith angle: these determine the relative importance of
higher-order scattering and geometry dependence of the solar source term in
the neighbourhood of the line of sight. The importance of higher-order
scattering is discussed in Sect.

Percent difference in simulated radiance between HR and
MC (

Seconds for MC to estimate the observed radiance for 3 wavelengths [

Table

HR simulations of the accuracy shown in Fig.

Direct comparison of the HR and SO (with coupled diffuse profiles) is
more straightforward. Runtime (per wavelength) to simulate radiance
over a large range of near-UV through near-IR wavelengths is shown in
Table

Representative runtime (per wavelength) and RAM usage for HR and SO for similar resolutions and various numbers of diffuse profiles (DP).

The SO engine was compared to several other radiative transfer models
in

HR and MC have been compared for a variety of solar conditions and
wavelengths. The atmosphere used is representative of a “standard”
atmosphere away from the Earth surface, consisting of Rayleigh
scatterers, ozone, and aerosol. The surface albedo was set to

Figure

Figure

Number of diffuse profiles needed to get 0.2 % agreement
with MC, at 10

As an example usage case, the two radiative transfer models are
applied to data from the Optical Spectrograph and InfraRed Imaging
System (OSIRIS), a limb-scatter instrument launched in 2001 on board
the Odin satellite

Typical movement of the Odin satellite (open circles) and tangent point (closed circles) as the line of sight is scanned down and up, shown in red and blue respectively. The bottom panel shows the ground tracks of the tangent points; contours mark lines of constant zenith angle.

Figure

A consequence of the scanning of the line of sight is that the line of
sight tangent point traverses a larger distance during down-scans than
up-scans, as up-scans tend to cancel the forward motion of the
satellite. This causes larger changes in the local illumination
conditions and has implications for the accurate modelling of the
limb-scatter radiances. The tangent point of an up-scan typically covers
approximately 4

Mean percent difference between retrieved ozone number
density when the forward model is run with one diffuse profile
compared to five, i.e.

From Figs.

Percent difference between retrieved ozone number density
when the forward model is run with one diffuse profile compared to
five, i.e.

To test this two studies were performed. First, approximately 2600
OSIRIS scans where it is difficult to accurately model the diffuse
field were selected from 2008 and 2009. These are scans with solar
zenith angles greater than 80

Mean percent error between the ozone profile retrieved when
using five diffuse profiles in the forward model and the simulated
known value, i.e.

Next a simulation study was performed where MC was used to simulate
the OSIRIS data with a SD of at most 0.2 %, roughly the reported precision
of OSIRIS radiance measurements in the UV. For simulation purposes
a monthly averaged ozone climatology, specified on a 500

Figure

Using more diffuse profiles in the retrieval forward model has the
effect of changing retrieved values by up to a few percent for solar
scattering angles far from 90

To better understand the effect as a function of altitude, we separate
scans into three distinct cases based on scattering angle,

In order to understand the cause of the bias, we need to understand how
changes in radiance affect the ozone retrieval. At high altitudes, the ozone retrieval uses
measurement vectors of the form

At low altitudes the opposite effect is observed. Here, the retrieval wavelength used is in the Chappuis band, with normalization wavelengths on both sides of the band. The relative sensitivity of these wavelengths to changes in the diffuse field depends on the amount and type of aerosol present. Overall, however, the retrieval wavelength is more sensitive to the diffuse field than the reference wavelengths, leading to an overestimation of the measurement vector and thus ozone.

Down-scans have the opposite effect of up-scans. For the same
geometry, the reference altitude measurement occurs at a solar zenith
angle less than the retrieval measurements. This means that the terms

Similarly, scans with solar scattering angle greater than 90

The primary advantage of retrieving from simulated measurements is
that the true state is known and can be compared against. In
Fig.

So far we have limited our discussion to ozone retrievals with OSIRIS
geometries; however similar effects should exist for other instruments
and species. The effect on other species is heavily dependent on the
exact retrieval algorithm used; thus we merely reiterate that when
using one diffuse profile the altitude normalized radiance,

Two new radiative transfer models have been developed within the SASKTRAN framework: A new high-resolution successive-orders model and a Monte Carlo reference model.

The high-resolution model is intended for use as an accurate spherical radiative transfer model that operates without the assumption of horizontal homogeneity of the atmosphere and is fast enough for use in limb-scatter retrievals. Regions of large extinction (e.g. cirrus clouds) are handled through an adaptive integration step. Variations in atmospheric composition along the horizontal direction are accounted for through new two- and three-dimensional atmosphere modes. Weighting functions for number density of scattering and absorbing species can be approximated analytically. These approximate weighting functions deliver better performance than those calculated using the traditional single-scattering approximation and require negligible time to compute compared to the full radiative transfer calculation.

The Monte Carlo model is intended for use as an accurate reference
model that estimates solutions to the radiative transfer problem
without bias. The model is implemented within the SASKTRAN framework
and is therefore useful as a tool for error checking other models
within the framework. Furthermore, it can been used to prescribe the
resolution necessary in faster successive-orders
discrete-ordinates
models to achieve accuracy to within some limit. In this work,
configurations were found that allow the high-resolution model to
agree with the Monte Carlo reference model to within

The two radiative transfer models were used to identify and eliminate
a bias in the OSIRIS ozone product. OSIRIS scans were simulated using
the Monte Carlo model, and vertical profiles of ozone were retrieved
from these simulated scans using the high-resolution model. It was
shown that calculating the multiply scattered diffuse radiance field
at only one solar zenith angle introduces a bias of up to

This work was supported by the Natural Sciences and Engineering Research Council (Canada) and the Canadian Space Agency. Odin is a Swedish-led satellite project funded jointly by Sweden (SNSB), Canada (CSA), France (CNES), and Finland (Tekes). Edited by: M. Weber