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Introduction to microwave radiative
transfer May 2002
By Peter
Bauer
European Centre for Medium-range Weather
Forecasts, Shinfield Park, Reading Berkshire RG2 9AX, United Kingdom
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1 . Radiative transfer
The radiative transfer equation can be expressed as the differential change of radiance
along path 
:
|
(1) |
In vertical coordinates, and including slant paths, the path coordinate is modified to optical depth
with zenith angle 
. 
denotes the volume
extinction coefficient comprising absorption and scattering by all relevant
media in the atmosphere. All optical quantities are frequency dependent,
thus the subscript `
' will be omitted
hereafter. Including source terms (1) translates to:
|
(2) |
denotes the azimuth angle so that the
angular dependence may be combined to 
, 
. The source term covers contributions from scattering (hydrometeors)
and emission (oxygen, water vapour, dry air, hydrometeors):
|
(3) |
Scattering of radiance is expressed in terms of a normalized scattering phase function:
|
(4) |
describing the distribution of incident radiance (
) to observation direction (
). 
denotes the single scattering albedo and provides a measure for the
fraction of scattered radiation while 
is the
fraction of absorbed / emitted radiation.
is the blackbody equivalent radiance according to temperature.
The differential form of the radiative transfer equation can be integrated from the surface (
) to the top of the atmosphere, 
:
|
(5) |
with total atmospheric transmission:
|
(6) |
and surface emissivity,
, which is a
function of temperature, roughness, foam coverage, and salinity in the case
of sea water and a function of temperature, moisture, soil type, vegetation,
and roughness (among others) for land surfaces. 1.1 Emission
In the case of a purely absorbing medium,
. Neglecting the azimuth angle dependence (6) becomes:
|
(7) |
The term
defines the atmospheric
transmission from level 
to the top of
the atmosphere, i.e.:
|
(8) |
which by insertion into (7) (neglecting surface contributions) provides the basic radiative transfer equation for vertical atmospheric sounding (note the exchange of the integration limits):
|
(9) |
In the microwave spectrum and in cloudfree areas, optical depth is mainly a function of atmospheric temperature (50-60 GHz, 118 GHz) and water vapour (22 GHz, 183 GHz) due to line absorption; however, continuum contributions by dry air and water vapour increase slowly with frequency and are non-negligible though difficult to parameterize.
The term
is called the
`weighting function' since it provides weights of the contribution of 
to 
. Thus only as much radiance from a certain
layer can reach the top of the atmosphere as is transmitted through the
overlying layers. Typically, the weighting functions have `Gaussian' shapes
with a maximum at the level where 
has a maximum gradient. This level defines the layer to which a sounding
has a maximum sensitivity; however the width of the weighting function indicates
that the sounding level is not discrete but rather blurred and depends on
the local conditions to be retrieved. For remote sensing applications, it is important to notice that the integrand in (9) comprises two profile variables, i.e.
and 
. An inversion of (9) is more accurate for temperature profile
retrievals near or at oxygen absorption lines because there, absorption
is a function of temperature as is 
. Since oxygen is a well mixed gas in the troposphere, absorption
does not depend on gas concentration but only temperature. In the case of water vapour profile retrievals,
is a function of water vapour content while 
is a function of temperature. Thus, both absorber density and temperature
profiles are convolved and cause less accurate retrievals of e.g. water
vapour contents at specific altitudes. Here, a combination of temperature
profile retrievals at, say 50-60 GHz, and water vapour profile retrievals
at 183 GHz are of advantage. In terms of water vapour content (e.g. mixing
ratio), the integrand in (9) may
be expressed as:
 .
|
(10) |
Here,
and 
are known so that (10) is as complex as for temperature sounding
but expressed in terms of the desired quantity. Of some importance is the way the integration in (9) is carried out. Considering a single layer, it may be assumed (1) that
with layer average temperature 
, or (2) that
changes linearly within the layer, i.e., 
with 
being the temperature at the top of the layer (where
) and 
being the lapse rate. Then:
|
(11) |
Depending on the optical depth of the layer, the difference between the two options may be several K. At microwave wavelengths, this becomes important in clouds and precipitation but may be neglected in optically thin cloudfree atmospheres.
1.2 Scattering
In the microwave part of the electromagnetic spectrum, the dependence of scattered radiation on azimuth angle can be neglected in most cases because multiple scattering of diffuse radiation is much less anisotropic than that of e.g. solar radiation. In this case, the phase function in (4) becomes:
|
(12) |
and (2) reduces to:
|
(13) |
A rather accurate approximation to the phase function in (12) is given by the Henyey-Greenstein function which is only applicable in scalar radiative transfer:
|
(14) |
denotes the asymmetry parameter which
represents the angle-averaged phase function, 
:
|
(15) |
This parameter is >0 / <0 if more radiation is scattered in forward/backward than backward/forward direction. For Rayleigh scattering
. 1.3 Polarization
Another important source of information is polarization, because surfaces polarize incoming unpolarized radiation by reflection and particles polarize by scattering. In both cases, a strong dependence of illumination vs. observation geometry exists, and the degree and angular distribution of the polarized radiation is determined by surface reflectivity and roughness, and particle scattering efficiency and shape. Polarization calculations require the expansion of radiances into vertically and horizontally polarized components which are elements of the Stokes vector
:
|
(16) |
The (v, h) are defined by a plane between the incoming and scattered/reflected radiation beams. `v' represents the vertical component and `h' the parallel component to this plane. Angle
defines the orientation of the vector 
with respect
to the `h'-direction while 
stands for the
ellipticity of the polarization. The sign of 
describes
the sense of rotation, i.e., the sign of the phase (
) difference
between 
and 
. Elliptically polarized radiation represents
the most general definition form for polarized radiation with special cases
of unpolarized, linearly polarized, and circularly polarized radiation:
thus 
, 
; 
, 
; and 
, 
, respectively. The consequence for the radiative transfer equation (13) is that all radiance terms become (Stokes) vectors, scattering and extinction coefficients become matrices, while the scattering phase function also becomes a matrix:
|
(17) |
is the Stokes vector representation
of the scalar intensity, 
is still scalar
because blackbody emission is unpolarized (
), 
denotes the 
scattering matrix (Mueller matrix), and 
has also become a 
matrix determining
the amount of scattering per component. The latter expansion is only required
if scattering by non-spherical particles is included, otherwise 
remains a scalar. Since scattering is always described in the local scattering geometry (as is the polarization), i.e., in reference to the plane determined by the incoming and scattered radiation beams with a particle at the center, a coordinate transformation has to be carried out before and after the scattering event with respect to (
):
|
(18) |
represents a rotation matrix:
|
(19) |
The most general form of the scattering matrix
is:
|
(20) |
which can be reduced to a more simple form for spherical particles:
|
(21) |
Other simplifications apply for particles with less general symmetry than spheres.
1.4 Boundaries
Downwelling radiation from space is commonly approximated by
(2.7 K) for all incidence angles. Surfaces can be
treated as reflectors which are specular for e.g. a calm water surface.
In that case, the Fresnel equations apply which translate to a reflection
matrix:
|
(22) |
with reflection coefficients for a medium with complex permittivity
:
|
(23) |
Surface transmissivity is neglected in most cases assuming a penetration depth
of zero so that
becomes a matrix in (5). Most natural surfaces, however, are rough
so that-at least theoretically-bistatic reflection coefficients have to
be calculated giving the fraction of scattered radiation for any incidence
and scattering angle combination. Another approximation to surface reflection
is represented by a Lambertian reflector for which the distribution of reflected
radiation is isotropic over all angles. 1.5 Antenna patterns
Microwave antennas on current satellites represent a compromise between desired spatial resolution at the surface and affordable antenna size. Since measurements at all frequencies are usually obtained with the same antenna, spatial resolution varies with frequency. Spatial resolution of the instantaneous field of view, IFOV, can roughly be estimated from (here at nadir):
|
(24) |
with satellite altitude
, antenna diameter
, an antenna efficiency factor 
(e.g. = 1.5)
and wavelength 
. This increases for inclined observation geometry
and distorts circular antenna patterns to ellipsoids. Since the antenna
size is more or less of the dimension of several wavelengths, diffraction
is important and leads to interference patterns. Most antennas have very
efficient main lobes but side lobe effects are mostly non-negligible over
scenes with strong horizontal gradients of radiance emission / scattering
(near coastlines and over clouds and precipitation). An idealization of
the antenna imaging is represented by the approximation of the main lobe
by a Gauss function with a halfwidth according to the nominal 3 dB beamwidth
given in technical documents (e.g. 40 km x 60 km for the SSM/I 19.35 GHz
channel). Thus a spatially inhomogeneous radiance field has to be integrated
over the radiometer field of view with the antenna pattern 
:
|
(25) |
The angle
represents the
effective field of view (EFOV) in coordinates of zenith and azimuth angles
and is the instantaneous field of view including its distortion by the scanning
motion of the antenna.
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12.06.2002 |
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