5.2 Radiative Transfer
5.2.1 Atmospheric Transmittance and Scattering.
The radiative transfer algorithms to be used for the OCO project were developed for the
radiative investigations of Earth and planetary atmospheres (Meadows and Crisp, 1996;
Crisp and Titov 1997; Crisp, 1997).
These methods include a pair of multi-level,
spectrum-resolving (line-by-line) multiple scattering models. The most accurate of these
is the Discrete Atmospheric Radiative Transfer (DART) model. This model creates a
spectral grid that is fine enough to completely resolve the spectral variability associated
with near-infrared line absorption and UV pre-dissociation and electronic bands of gases,
as well as the wavelength dependence of the optical properties of airborne particles and
the surface. This brute-force numerical model then performs a monochromatic multiple
scattering calculation at each spectral grid point to produce a wavelength-dependent solar
spectrum. The Spectral Mapping Atmospheric Radiative Transfer (SMART) model also
explicitly resolves the wavelength and altitude dependence of the atmospheric and
surface optical properties, but this model employs high-resolution spectral mapping
methods (c.f. Crisp and West, 1992; Meadows and Crisp, 1996, Crisp, 1998) to minimize
the number of monochromatic multiple scattering calculations needed to generate high
resolution synthetic spectra in broad spectral regions.
The spectrum- and altitude-
dependent atmospheric gas absorption coefficients required by DART and SMART are
obtained from a state-of-the-art line-by-line model (LBLABC, Meadows and Crisp
1996).
The wavelength-dependent, single-scattering optical properties for spherical
liquid water droplets are obtained from a Mie scattering model, and those for hexagonal
water ice crystals are derived from a model that incorporates geometric optics and a
modified Kirchhoff approximation. A high-resolution solar spectrum (Kurucz, 1995) will
be used for the simulations proposed here. These methods are described in greater detail
below.
Spectrum-Resolving Multiple Scattering Models:
Both DART and SMART use the multi-level, multi-stream, discrete ordinate algorithm,
DISORT (Stamnes et al. 1988) to generate altitude- and angle-dependent solar radiances
at each wavelength of interest in plane parallel, scattering, absorbing, emitting
atmospheres.
DISORT was chosen for this application because of its speed and
accuracy, and because a well-documented, numerically-stable code was readily available
(ftp://climate.gsfc.nasa.gov/pub/wiscombe/Multiple_Scatt/). In this model, the angular
integrals in the equation of transfer are evaluated using Gaussian quadrature to yield
radiances at a specified number (typically 2 to 32) of discrete zenith and azimuth angles
for each atmospheric layer. Vertical inhomogeneity is accounted for by dividing the
atmosphere into a series of homogeneous layers. The optical properties (optical depth,
uniform throughout each layer, but these properties can vary from layer to layer. When 4
or more streams are used, these methods usually produce angle dependent
monochromatic radiance errors no larger than 1% for clear, cloudy, and aerosol-laden
atmospheres, but they will produce somewhat larger errors (10%) in atmospheres where
Rayleigh scattering dominates because they neglect the effects of polarization (Adams
and Kattawar, 1978; Mishchenko et al. 1994).
DISORT, like most monochromatic multiple scattering algorithms, can provide accurate
solutions to the equation of transfer only when it is applied in spectral regions that are
sufficiently narrow that the optical properties and source functions are roughly constant
across each. The two radiative transfer models used in this investigation employ different
approaches for resolving the wavelength dependence of the surface and atmospheric
optical properties and the solar source function. The DART model divides the solar
spectrum into a numerical grid that is sufficiently fine to completely resolve the
wavelength dependence of all radiatively-active constituents at all points along the
optical path. The approach used to define the optimum spectral grid spacing is described
in Meadows and Crisp (1996). DART then uses DISORT to perform a monochromatic
multiple scattering calculation at each spectral grid point. Such methods are often called
line-by-line multiple scattering models, by analogy to the line-by-line transmission
models that are used for clear-sky transmission calculations at infrared wavelengths (cf.
Fels and Schwarzkopf 1980; Clough et al. 1989). However, the term spectrum-resolving
multiple scattering models is somewhat more appropriate because these models must
resolve the wavelength dependence of UV electronic transitions of gases, the single
scattering optical properties of cloud and aerosol particles, and surface albedos, as well as
near-infrared gas absorption lines.
Direct numerical methods like DART should produce the most reliable results possible
because they can use all of the available information about the atmospheric and surface
optical properties, and employ a minimum number of approximations. Their principal
drawback is their large computational expense. For example, for broad-band solar flux
calculations, DART resolves the atmospheric and surface optical properties into about 3
million unequally spaced spectral points at wavelengths between 0.125 and 8 microns.
About 90 % of these points are required at near-infrared wavelengths (0.6 to 8 microns)
where gas vibration-rotation transitions contribute to the spectral variability. A multilevel, multi-stream monochromatic multiple scattering calculation must then be
performed at each spectral point. For model atmospheres with 60-65 vertical levels, a 4stream DART calculation usually requires 1 to 2 days to produce radiances, fluxes, and
heating rates throughout the solar spectrum for a single solar zenith angle on a high
performance desk-top workstation. Hence, even though these methods provide valuable
standards for comparison, they are still impractical for routine use.
The Spectral Mapping Atmospheric Radiative Transfer (SMART) model employs the
same input data, the same spectral grid, and the same multiple scattering algorithm used
by DART, but this model uses high resolution spectral mapping methods to enhance its
computational speed (c.f. Crisp and West, 1992; Meadows and Crisp, 1996; Crisp, 1997).
Like the correlated-k (c-k) method (Lacis et al., 1979; Lacis and Oinas, 1989; Goody et
al. 1988; Fu and Liou, 1992), and the optical depth binning method introduced by
Ramaswamy and Freidenreich (1991), spectral mapping methods gain their efficiency by
identifying monochromatic spectral intervals that have similar optical properties. These
intervals are then gathered into bins, and a single monochromatic multiple scattering
calculation is performed for each bin. However, the binning methods used in the spectral
mapping algorithm differ from those used in the other two methods. In particular, both
the c-k methods and the optical depth binning method assume that atmospheric optical
properties are spectrally correlated at all points along the optical path, such that
monochromatic intervals with similar optical properties at one level of the atmosphere
will also remain similar at all other levels. If this condition holds, a monotonic ordering
of the absorption coefficients (or optical depths) performed at any pressure level is
exactly preserved at all other levels.
This assumption is rigorously valid for
homogeneous, isobaric, isothermal optical paths, but it usually breaks down for realistic
inhomogeneous, non-isothermal, atmospheric optical paths. This loss of correlation can
sometimes introduce significant flux and heating rate errors (c.f. Goody et al. 1989, West
et al. 1990; Fu and Liou, 1992).
In contrast, spectral mapping methods make no
assumption about the spectral correlation along the optical path. Instead, these methods
perform a level-by-level comparison of monochromatic atmospheric and surface optical
properties, and combine only those spectral regions that actually remain in agreement at
all points along the inhomogeneous optical path. The spectral mapping approach is not
as efficient as these other methods, but it is usually more reliable for use in
inhomogeneous, non-isothermal atmospheres because it specifically avoids errors
associated with the loss of correlation along the optical path.
SMART generates a high-resolution, angle-dependent solar radiance spectrum through
the following series of steps. Like DART, it first defines the composite (gas and particle)
optical depth,
each atmospheric layer, at each spectral grid point in a multi-layer, scattering, absorbing
atmosphere (c.f. Liou et al. 1978). Surface albedos and bi-directional reflection
functions are specified as the lower boundary of the model, and solar fluxes are specified
at the top of the atmosphere at each spectral grid point. Next, the spectral mapping
algorithm employs a user-defined binning criteria to identify all spectral grid points that
have optical properties that remain similar at all levels of the atmosphere and at the
surface (see Method 2 in West et al. 1990).
Similar monochromatic intervals are
collected into bins. Unlike the broad-band spectral mapping methods introduced by West
et al. (1990), SMART then records the bin number associated with each grid point in a
spectral map that is later used to map the computed radiances back to a full-resolution
spectral grid (see Meadows and Crisp, 1996). Once similar spectral intervals have been
gathered into bins, the mean optical properties for the bin are computed (c.f. West et al.,
1990), and DISORT is used to perform a single monochromatic multiple scattering
calculation for each bin (assuming a unit solar flux and/or no solar flux at the top of the
atmosphere).
For the calculations proposed here, monochromatic radiances will be
obtained for 4 to 32 zenith angles (or streams) and 7 to 16 azimuth angles at 60 to 65
levels between the surface and 80 km for each solar zenith angle of interest. The leveldependent radiances for each bin are then mapped back to their original spectral grid
points, and multiplied by the solar flux at that wavelength to produce a high spectralresolution, angle-dependent description of the radiation field at each atmospheric level.
For solar calculations like those described in Crisp (1997, 1998), SMART combined the
6 million discrete spectral segments between 0.125 and 8 microns into about 360,000
unique spectral bins. The radiances for each bin were then computed, and mapped back
to their original spectral positions, and convolved with a 2 cm-1 wide (full-width at half
maximum) triangular slit function for presentation, or integrated over angle and
wavenumber to yield broadband solar fluxes and heating rates. Comparisons between
SMART and DART indicate that even though spectral mapping methods can reduce the
number of monochromatic calculations needed in broad spectral intervals by about a
factor of about 100, they rarely introduce radiance errors larger than 2% in spectral
intervals wider than cm-1. SMART therefore provides the accuracy and efficiency needed
even for global scale calculations, but it still does not provide the computational speed
needed for routine use in GCMs.
The Line-by-Line Model for Gas Absorption:
DART and SMART require a comprehensive, wavelength-dependent description of the
absorption by gases at each atmospheric level throughout the desired spectral range.
Here, we will derive monochromatic gas absorption coefficients from a state of the art
line-by-line model called LBLABC (see Meadows and Crisp, 1996). LBLABC provides
accurate estimates of the absorption coefficients over a very broad range of pressures,
temperatures, and line-center distances (10-3 to 103 cm-1).
LBLABC was therefore
designed to evaluate gas absorption coefficients at each atmospheric level on a series of
nested spectral grids. The spacing of the finest grid is chosen to completely resolve the
cores of the narrowest absorption lines (4 to 8 points per Doppler half-width). Because
the rate of change of the line shape profile decreases with increasing distance from the
line center, the spacing of each successive grid increases by a factor of exp{1}. Up to 20
nested grids can be used to provide a range of grid resolutions exceeding 108. The
absorption coefficients for each line are evaluated at only 10 to 20 positions on each grid.
This multi-grid approach requires between 100 and 200 evaluations of the line-shape
function for each spectral line, even when this line contributes significant absorption at
distances as large as 1000 cm-1 from the line center. Once the absorption coefficients
have been obtained for all lines that contribute to the spectral interval of interest,
contributions from each grid are interpolated to a single high-resolution output grid,
summed and then saved to disk for later use.
LBLABC models the line shape function differently in the line-center and far-wing
regions. For line-center distances less than 40 Doppler halfwidths, a Rautian line shape is
used.
This line shape incorporates Doppler broadening, collisional (Lorentzian)
broadening, and collisional (Dicke) narrowing.
At greater distances, a van Vleck-
Weisskopf profile is used for all gases except for H2O, CO2, and CO (c.f. Goody and
Yung, 1989). The super-Lorentzian behavior of the far wings of H2O lines, which has
been attributed to the finite duration of collisions, is parameterized by multiplying the
Van Vleck-Weisskopf profile by a wavenumber-dependent X factor.
The X factor
recommended by Clough et al. (1989) has been adopted in most of our published work,
and will be used for the simulations proposed here. This profile provides significant
absorption at line-center distances exceeding several hundred wavenumbers.
This
absorption is adequate to account for the water vapor continuum absorption seen at
thermal infrared wavelengths, and even produces a weak continuum throughout much of
the visible and near-infrared spectrum (Crisp 1997, 1998).
The CO2 line profile is affected much less by the finite duration of collisions. In fact,
low-resolution laboratory measurements indicate that the far wings of CO2 bands can be
more accurately simulated by assuming a sub-Lorentzian line profile (Burch et al. 1969).
A large body of recent theoretical work (c.f. Levy, 1992 and references therein) has now
confirmed that this behavior is primarily a consequence of a vibration-rotation energy
redistribution process called collisional line mixing. This process is most effective when
pressures are high enough that the line half-widths are comparable to the line spacing.
Under these conditions, collisions between molecules can transfer energy between
rotational states of a band. The net effect of this process is usually to transfer energy
from higher to lower rotational energy levels. Line mixing has no direct effect on the line
profile, but it causes an apparent narrowing of the band (as seen in low-resolution
spectra), and reduced absorption at large distances from the band center. In the terrestrial
atmosphere, this process is most pronounced in the vicinity of the narrow Q-branches of
the CO2 15 micron band, where lines are spaced at distances that are small compared to
their widths even at stratospheric levels (c.f. Strow and Reuter, 1988). At tropospheric
pressures, line mixing affects even the much more widely spaced CO2 P and R branches.
Direct ab initio methods now exist for computing the effects of line mixing (c.f.
Hartmann and Boulet, 1991), but these methods do not yet provide the accuracy or
numerical efficiency required for routine use. LBLABC currently employs a simple,
semi-empirical algorithm that attempts to correct for the effects of line mixing at very
low pressures (Rosenkranz, 1988) and very high pressures (Meadows and Crisp, 1996).
These methods should provide the accuracy needed for the calculations proposed here,
but more rigorous methods are continuously being sought to support our ongoing
planetary atmosphere modeling tasks.
Single Scattering Optical Properties of Cloud and Aerosol Particles:
To provide an accurate description of the solar and thermal radiation fields in cloudy or
aerosol laden atmospheres, spectrum-resolving multiple scattering models like SMART
and DART require a detailed, wavelength dependent description of the single scattering
optical properties of the cloud and aerosol particles. This information is essential both
for the retrieval of cloud and aerosol properties from remote sensing observations, and to
estimate their contributions to the solar and thermal radiative forcing in climate models.
Mie scattering algorithms can provide reliable estimates of these properties for spherical
cloud liquid water droplets or sulfuric acid aerosol droplets, but these methods are not
valid for non-spherical particles like those that compose cirrus clouds, or many common
aerosol types (soot, desert dust, sea salt, etc.). The common practice of modeling these
particles as equivalent volume spheres can produce errors as large as several hundred
percent (Chylek and Videen, 1994).
Several methods have been developed to find the optical properties of non-spherical
particles, but each has its limitations.
For example, analytical models have been
developed for non-spherical particles that are very small compared to the wavelength
(van de Hulst, 1957), while Geometric Optics methods (c.f. Takano and Liou, 1989) are
only valid for particle sizes much larger than the wavelength. Anomalous Diffraction
Theory can produce absorption and scattering cross sections for distributions of
hexagonal plates and prisms of arbitrary size, but this method provides no information
about the scattering phase function, and is only valid when the particle refractive index is
close to unity (Chylek and Videen, 1994). The Discrete Dipole Approximation (Purcell
and Pennypacker, 1973; Draine, 1988) can be used to calculate the scattering from an
arbitrary object by replacing it with an array of elementary electric dipoles, thus utilizing
only the electric dipole moments of a source distribution. As the size parameter of the
scatterer increases, other contributions in addition to the electric dipole moments must be
considered. This method has been used to calculate the optical properties of aggregate
particles with sizes comparable to the wavelength (West, 1991), but concerns about
accuracy and computational speed limit its utility for larger particles. Another class of
methods based on integral equation (full wave) techniques has recently been developed
(Ishimaru, 1991).
In principle, this approach should be exact, without any explicit
frequency limitations. However, general three-dimensional volumetric integral equations
are difficult to implement. A Surface Integral Equation formulation called the Extended
Boundary Condition (T-matrix) method provides an efficient, accurate approach for
modeling the optical properties of spheroids and cylinders (Mishchenko, 1991), but this
method is not practical for particles with sharp corners or large axial ratios.
Zuffada and Crisp (1997) developed an alternate surface integral approach for deriving
the optical properties of axisymmetric particles with large axial ratios.
They
implemented this method in program called BOR-IE, and demonstrated its accuracy for
spheres and spheroids with sizes spanning the Rayleigh and geometric optics regimes.
They also derived the wavelength dependent optical properties for water ice cylinders and
disks. They found ice particles with these shapes were more strongly absorbing than
equivalent volume ice spheres at most infrared wavelengths. They also found that the
geometry of the particles strongly affects the angular dependence of the scattering. In
particular, disks and columns with maximum dimensions that are comparable to or larger
than the wavelength scatter much more radiation into narrow forward and backward
peaks, and much less radiation into intermediate phase angles than equivalent volume
spheres.
These results have important implications for the measurment techniques
needed to quantify the abundance, composition, and physical properties of non-spherical
particles from ground-based and space-based platforms.
The principal limitation of the BOR-IE model is its restriction to axially-symmetric
particles. Methods for extending its capabilities to handle particles of arbitrary shape will
be pursued.
Specifically, we will assess the accuracy of a finite-element-integral-
equation technique that should be ideally suited for inhomogeneous particles large
compared to the wavelengths of interest (Zuffada et al. 1997, Cwik et al. 1996). While
this approach promises to enhance our capabilities for modeling the scattering by
particles in the resonance regime (i.e. where the wavelength is comparable to the particle
size), we do not propose that it is ideal for all non-spherical particle calculations. Instead,
we propose to incorporate this scheme with our existing suite of single scattering particle
codes, including Mie scattering models (Wiscombe, 1980), geometric optics/modifiedKirchhoff models (Muinonen et al. 1989), and T-matrix models (Mishchenko, 1991) to
produce an accurate description of the particle optical properties over the full range of
sizes, shapes, and wavelengths.
5.2.2 Surface Reflectance
The bidirectional reflectance distribution function (BRDF) quantifies the angular
distribution of radiance reflected by an illuminated surface. We will discuss sea and land
surfaces separately.
Sunglint
If the sea surface were perfectly flat, a single disk-like image of the sun would be seen at
the specular reflection angle. However, this never happens in reality because the sea
surface is always rough due to wind driven waves. In the classic study of sunglint from a
sea surface, Cox and Munk [1954a, 1954b, 1955] realized that the width of the glint’s
pattern is an indication of the maximum slope of the sea surafce. The key parameters that
characterize the surface reflection characteristics are the wind speed and direction, which
may be estimated from assimilation models, such as the ECMWF Reanalysis.
Land Surface
The BRDF for land surfaces has recently been classified as 17 biome types on a
0.10.1 grid for visible to near infrared wavelengths [Roujean et al., 1997;Maignan et
al., 2004 ]. The BRDF model takes the form of
R(s, v, ) = k0 +  kiFi(s, v, )
where s, v and  are the solar zenith, view zenith and relative azimithal angles,
respectively, Fi are empirical functions derived from observed data, and ki are free
parameters. The current model has three parameters (i.e., k0, k1 and k2).
The model uses data from POLDER, which has channels at 443, 670 and 865 nm.
The coefficients are available to this group. To cover the wavelength range required by
OCO, we will extend the BRDF using MODIS data. In general, the bidirectional
signatures get larger as the reflectance gets smaller. We will use k0 from MODIS, but
adopt values of k1/k0 and k2/k0 similar to those at shorter wavelengths. Work is in
progress to characterize BRDF of snow and ice surfaces.
5.2.3 Computing Partial Derivatives
We have identified three approaches to computing partial derivatives.
Brute Force
The radiance is calculated for the base state and then for the perturbed state, and the
jacobian is computed using a finite difference technique.
K
F ( x, x' ) F ( x  x, x' )  F ( x, x' )

x
x
where K is the jacobian, F is the radiance, x is the parameter being perturbed, x is the
perturbation step size and x’ refers to the other parameters influencing the radiative
transfer.
The problem with this technique is twofold. Firstly, the RT model has to be called
repeatedly for calculating the partial derivatives. Secondly, a lot of redundant
computation is performed every time, since in general for any perturbation only a
particular atmospheric layer is affected. However, the method employed above does not
take advantage of this.
Layer Saving Mode
Here, we still use finite differences. However, usage of the doubling-adding technique
allows saving of the reflection and transmission matrices for each layer when the base
state is computed. For the perturbed state, the transmission and reflection matrices are
recomputed only for the particular layer that is affected. The saved values from the base
state are used for the other layers, and the global reflection and transmission matrices are
computed very efficiently.
For parameters in the state vector which are not layer dependent (e.g., surface albedo and
surface pressure), it would appear that this technique would not work. However, this is
not a problem. For the surface albedo, the entire atmosphere from the unperturbed state is
used and the new surface albedo boundary condition applied internally to obtain the new
radiance. In the case of surface pressure, when the surface pressure changes, then the
pressure in every atmospheric layer will change (through the hydrostatic equation).
Because of this, all the gas absorption coefficients change through their pressure
dependence. For this case alone, the full calculation needs to be done for the perturbed
state.
Analytic Jacobian
Both the above methods require multiple calls to the RT model to obtain the jacobian.
However, an internal perturbation analysis could be done to simultaneously compute the
weighting functions along with the base state radiances. Clearly, this would tremendously
reduce computation time, since the evaluation of the jacobian in the brute force method
takes more than 80% of the total CPU time. In addition, this technique also guarantees
numerical stability since it does not depend on the perturbation step size. A radiative
transfer package called LIDORT has been developed to compute the jacobian
analytically, and we hope to implement it in our retrieval algorithm in the near future.