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Inversion methods for satellite
sounding data
April 1991
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1 . Basic ideas
1.1 The radiative transfer equation for emission to space
The monochromatic radiation intensity at frequency
emitted along a vertical path at the top of the atmosphere
and incident at a satellite-borne instrument is given by:
|
(1) |
where
 is the emission
from the earth's surface at height  , |
 is the vertical
transmittance from height  to space, |
 is the vertical
temperature profile, |
and  is the corresponding
Planck function profile. |
Here we have neglected molecular scattering in and out of the beam - a good approximation in the infrared and microwave regions. We have also assumed, for the moment, that no cloud is present and we shall return to the cloud problem later. If the earth's surface reflects radiation significantly, then we acquire a third term in (1) representing radiation emitted downwards by the atmosphere and reflected back in the direction of the satellite. For simplicity we have ignored this term, which is equivalent to assuming that the surface is black (often a good approximation in the infrared). Reflection of solar radiation by the surface may usually be neglected also.
Equation (1) may also be used to represent radiation emitted along a slant (non-vertical) path if the transmittance is computed appropriately. Making the approximation of a plane-parallel atmosphere, for a viewing path through the atmosphere at angle
to the vertical, then
 ,
|
(2a) |
where
, 
, 
are respectively the vertical profiles
of atmospheric density, absorbing gas mixing ratio and absorption coefficient.It is often more convenient to choose pressure as the vertical coordinate. Then, using the hydrostatic approximation (
), we obtain
|
(2b) |
1.2 Integration over frequency
Real satellite instruments sense radiation over a range of frequency rather than monochromatic radiation, and it is usually necessary to perform an integration over frequency to obtain radiances of adequate accuracy as "seen" by the satellite instrument, i.e.
 ,
|
(3) |
where
is the relative response of the instrument to radiation at
frequency 
. This complicates the calculations involved
in the interpretation of the data, but does not change the basic nature
of the inversion problem. Therefore, for this discussion, we shall ignore
it and work only with the monochromatic equations.1.3 Weighting functions
Equation (1) may be written as
 .
|
(4) |
is called a WEIGHTING FUNCTION; it weights the Planck function in
the atmospheric component of the emitted radiation. It specifies the layer
from which the radiation emitted to space originates, and hence it determines
the region of the atmosphere which can be sensed from space at this frequency.
Fig. 1 shows the transmittance profiles and
corresponding weighting functions at two frequencies for which the atmospheric
absorption is different. Since the weighting function is the derivative
of the transmittance profile, it will peak higher in the atmosphere for
the frequency at which the absorption is stronger. In this way, a carefully
selected family of frequencies can be chosen to sense radiation from different
layers in the atmosphere.
Figure 1 . Idealised transmittance profiles and weighting functions at two frequencies with different absorption coefficients. Vertical coordinate: scale height = - loge(pressure).
To understand qualitatively why the weighting functions take this form, we can consider the emission to space from air parcels of unit volume at different heights in the atmosphere. The radiation emitted to space is determined by three factors:
| (a) the temperature of the air parcel, i.e. the variable we hope to measure, |
| (b) the number of molecules of emitting gas, which is determined by the atmospheric density (and also by the mixing ratio of the absorbing constituent, although for the principal gases used in temperature sounding-carbon dioxide and oxygen-the mixing ratio can be assumed constant and known), |
| (c) the transmittance of the atmosphere from the air parcel to space. |
This is illustrated in Fig. 2 for three air parcels at different heights. For the lowest parcel, the atmospheric density is high and so the amount of radiation emitted is high, but almost all is absorbed by the atmosphere above it and very little reaches space. For the highest parcel, the transmittance to space is high, but comparatively little radiation is emitted because atmospheric density decreases exponentially with height. These two conflicting effects combine in such a way that, at some intermediate height, the contribution of a parcel to the radiation reaching space is a maximum. The variation of the radiance to space as a function of height is shown by the curve on the right of Fig. 2 . Most of the radiation to space originates in a layer around the peak of this function (which is actually the product of the weighting function and the Planck function profile, i.e. the integrand in equation (4)). From knowledge of the atmosphere's composition and spectroscopic parameters we can calculate where in the atmosphere this layer will be. Then the intensity of the radiation can be interpreted in terms of the mean temperature of the layer. Using radiation at different frequencies for which the absorption strength is different, we can build a family of weighting functions, which provide information on the mean temperatures of many such layers, thus leading to the idea that we might be able to RETRIEVE information on the atmospheric temperature profile from a set of multi-frequency measurements.1
Figure 2 . Left-illustrating the attenuation of upwelling radiation emitted from three heights in the atmosphere. Right-the corresponding vertical profile of the contribution to the emission to space.
1.4 Characteristics of weighting functions
At this point, two aspects of the problem are worthy of note. Firstly, the weighting functions are broad (i.e. several kilometres). This means that the satellite instrument can sense the mean properties of broad layers very well, but it is only able to sense the characteristics of single levels or narrow layers insofar as they are correlated with the properties of the broad layers. The width of the weighting functions severely limits the capability of satellite sounders to detect atmospheric structure which has relatively small scale in the vertical. The finite width of the weighting functions is a fundamental feature of passive remote sensing techniques. However, the precise width is determined by technological considerations, as explained below.
Secondly, for most instruments, the family of weighting functions are highly overlapping. One consequence of this is that, although the instrument may make measurements at N separate frequencies, we obtain fewer than
pieces of independent information. The implications of this in the
inversion problem are discussed below.
Figure 3 . TOVS normalised weighting functions (from Smith et al. 1979).
Figure 3 illustrates the weighting functions for the present operational sounding system-the TIROS-N Operational Vertical Sounder (TOVS) which consists of 3 instruments: the High-resolution Infrared Radiation Sounder (HIRS/2), the Microwave Sounding Unit (MSU) and the Stratospheric Sounding Unit (SSU). For further information on TOVS, see Smith et al. (1979) or Schwalb (1978).
For the microwave channels, the spectral responses at the individual measurement frequencies (usually called "channels") are much narrower than the widths of the atmospheric absorption lines. Therefore the weighting functions are close to their monochromatic limit. However, it is possible to improve the vertical resolution of the microwave sounder by adding more channels. This will be done for the Advanced Microwave Sounding Unit (AMSU) which, along with HIRS, will constitute the Advanced TOVS on the next generation of polar orbiting satellites (from about 1994). The weighting functions for AMSU are illustrated in Fig. 4 .
Figure 4 . AMSU normalised weighting functions.
For the infrared channels the position is very different: HIRS is a filter radiometer and its channels have spectral widths typically hundreds of times greater than the atmospheric absorption lines. Therefore they average together frequencies for which the absorption strengths are very different. This has the effect of broadening the weighting functions considerably. By using instruments of much higher spectral resolution, such as interferometers or grating spectrometers, it is possible to achieve spectral resolutions closer to the widths of the atmospheric absorption lines. In this way instruments with several thousand channels and much sharper weighting functions can be built. Fig. 5 illustrates the weighting functions from such an instrument. Similar instruments are planned for satellite missions in the late 1990s.
Figure 5 . A selection of weighting functions for several bands of the interferometer proposed for the Atmospheric InfraRed Sounder (AIRS) instrument.
1.5 The forward and inverse problems
The instrument makes measurements of radiance in a number of channels
. For each channel, we can write a radiative
transfer equation:
|
(5) |
This equation expresses the FORWARD PROBLEM for the channel, i.e. given the state of the atmosphere, the solution of this equation tells us the radiance incident at the satellite in this channel. However, when presented with satellite measurements, we are faced with the INVERSE PROBLEM: given the measurements, what is the state of the atmosphere (in terms of its vertical profiles of temperature and constituents). Let us concentrate on the temperature profile inversion problem and return to the constituent problem later.
Since we have a limited number of channels
, we can see immediately that the inversion of equation (5)
is ILL-POSED or UNDER-CONSTRAINED. This is because we are trying to retrieve
, a continuous function of height (which, in general, requires
an infinite number of parameters to represent it fully), from a finite number
of measurements. This means that there exists an infinite number of profiles
which satisfy the measurements. Our problem is to find one which
is reasonable and, if possible, to find the profile which is best or most
reasonable in some sense.In addition, the measurements always contain some error or "noise". This further increases the ill-posed nature of the problem, and we must find a method of solution which does not amplify the noise to an unacceptable degree.
1.6 A vector-matrix representation
At this point, it is convenient to change from the notation of continuous profiles and integrals, as in equation (5), to discrete profiles and the notation of vectors and matrices. We consider the atmosphere to be composed of many thin layers (numbered, from the top,
) with mean temperature 
and Planck
function 
. Let the transmittance from the bottom of layer
to space be denoted 
. Then equation
(5) becomes
|
|
|
(6) |
 .
|
To solve this equation for
, it is convenient
to find a transformation of 
which is independent of 
(i.e. of frequency). For channels which are very close together in
frequency, we can use the Planck function at a central frequency. However,
this is rarely an adequately accurate approximation. One solution is to
specify a reference frequency for the Planck function and "adjust" all the
measured radiances to it. Alternatively, we can convert radiances to some
other quantity, such as equivalent black body temperature, which is independent
of frequency. These are technical details on which we need not dwell; it
is only necessary to appreciate that it is relatively straightforward to
find a channel-independent form of 
so that we may write
 .
|
(7) |
We can also "absorb" the surface term as the
th term in the summation by setting 
and 
. Then
 .
|
(0a) |
If we now represent the radiance in all channels by a vector
and the Planck function profile by a vector 
, equation (7) may be written for all channels simultaneously
as
 .
|
(8) |
is a matrix containing discrete weighting function
elements 
. Thus, our measurements are a vector
(elements 
), our unknowns are a vector 
(elements 
), and 
is a matrix of size 
. Our problem is to invert equation (8) to find 
. Then the temperature profile is obtained directly as a
known function of 
.1.7 Linearity
It is usually important to appreciate the degree of LINEARITY of any given inverse problem. By this we mean the degree to which we can separate out the knowns and unknowns of the problem into a linear equation. For example, in the case of equation (8), it represents a linear problem if
is independent of 
.
If 
is a function of 
, we have a nonlinear problem. In the temperature
retrieval problem, 
consists of differences between transmittances to space from the top and
bottom of the layer (see equation (6)). The transmittances, when expressed
in pressure co-ordinates (see equation (2b)), are strong functions of the mixing
ratio and pressure of the absorbing gas and its spectroscopic parameters,
but the latter are only weakly temperature dependent. Therefore, the problem
is almost linear. This means that we can calculate a reasonable approximation
to 
without accurate prior knowledge of the unknown 
. It also means that the weighting functions
for a given channel are almost independent of the atmospheric conditions.The near linear nature of the temperature inversion problem has allowed the development of appropriate inversion methods based on linear theory. Nevertheless, the nonlinearities are significant and must be considered carefully when accurate results are required.
An excellent discussion of linear inversion theory applicable to a wide range of geophysical problems is given by Menke (1984).
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1 We note in passing that ground-based measurements of downwelling atmospheric radiation do not have associated with them weighting functions of the same form. Here, the atmospheric density and transmittance from air parcel to instrument both decrease with height, and so (for an absorber mixing ratio which is constant with height) the largest contribution to the measured radiance is always from close to the instrument, whatever the frequency.
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07.06.2002 |
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