Submitted to applied optics, June 2002
Volume Emission Rate Tomography From a Satellite Platform
Douglas A. Degenstein, Edward J. Llewellyn and Nicholas D. Lloyd
Institute of Space and Atmospheric Studies
Department of Physics and Engineering Physics
116 Science Place
University of Saskatchewan
Saskatoon, SK, S7N 5E2
Canada
Received ______________; Revised ______________; Accepted ______________
Abstract
The possibility of retrieving horizontal atmospheric structure from a series of limb images taken
aboard a satellite is discussed and a Maximum Likelihood expectation Maximization (MLEM)
algorithm is developed. Examples of the retrieval of horizontal structure with this algorithm, for
different S/N ratios and different structures, are presented. It is shown that with this algorithm
and even in the presence of substantial observational noise, S/N equal to 10 for a single
observation, it is possible to retrieve both horizontal and vertical atmospheric structure.
© 2001 Optical Society of America
OCIS codes: 010.1280, 070.5010, 100.695, 110.6960
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Submitted to applied optics, June 2002
Introduction
Limb imaging of the atmosphere from space offers an opportunity to extract height profile
information for either the airglow emissions or the Rayleigh scattered sunlight. However, the
traditionally adopted retrieval techniques are usually required to assume some form of horizontal
homogeneity. While this assumption has worked reasonably well, although it can lead to
erroneous retrievals, it necessarily fails to recover horizontal structure. Thus the idea of making
some form of tomographic retrieval has acquired significant interest within the aeronomy
community as both the OSIRIS instrument on the Odin satellite [1] and the SCIAMACHY
instrument [2] on the Envisat-1 satellite make atmospheric limb observations. Previous work on
atmospheric tomography has been reported by Solomon et al. [3, 4] and Yee et al. [5] for
observations made with the visible airglow photometer on the AE satellite, by McDade et al. [6]
and McDade and Llewellyn [7] for rocket observations of an aurora, and by McDade and
Llewellyn [8] and Frey et al. [9] for a set of simulated satellite observations. In this paper we
present a new analysis with a tomographic algorithm and show that in addition to retrieving
height profile information it is possible to detect and retrieve horizontal structure with a
wavelength as small as 100 km even in the presence of substantial observational noise.
The Tomographic Algorithm
The de-blurrring of Fabry-Perot images has been considered by Lloyd and Llewellyn [10]
presented the problem as a linear system of equations, B  AT , where the vector B (indexed by
i) is the measured image of the discrete object T (indexed by j) which is blurred, or distorted, by
the apparatus dependent matrix A. Lloyd and Llewellyn developed a technique, the Maximum
Probability (MP) method, that considered the most probable contribution of any element j to the
measurement Bi, given that the Tj values are distributed according to photon counting, or
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Poisson, statistics. They developed the concept that each Tj value is the mean value, if the
contribution of that object element to the image is Pij, and in the mean Pij = AijTj. In their
analysis Lloyd and Llewellyn [10] used a Lagrange multiplier approach to develop equations (1)
and (2)
Pij 
B
i
 ni  Aij T j
A T
ij
1
(1)
j
j
P

A
ij
Tj
i
(2)
ij
i
where Pij is the most probable value of the contribution to the measurement Bi, Tj is the mean
value of the object and ni is the number of elements that contribute to measurement i. As these
equations are iterative equation (1) can be applied to an initial estimate of the T object elements
to yield a set of Pij values that may be used with equation (2) to give a new set of object
elements. The process is repeated until convergence is achieved. Lloyd and Llewellyn [10] also
demonstrated that a system of linear equations O  LV , where O is a set of zenith looking
rocket photometer data, V is a vertical airglow profile and L is a set of geometry and instrument
dependent weights, could be solved in a similar manner in order to determine the vertical airglow
profile.
In a subsequent development McDade et al. [6] were able to extend the MP technique to recover
the two dimensional N 2 volume emission rate profile in an aurora from rocket photometer data.
The matrix L was calculated from a knowledge of the rocket attitude and the measuring
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instrument characteristics. This L matrix also included the albedo contributions to the measured
signals.
In an extension of their work McDade and Llewellyn [7] simulated other rocket experiments that
considered a non-stationary auroral form. These same authors [8] also developed a simulation of
photometric measurements made with a simple vertical imaging instrument on a satellite
platform. They demonstrated that the tomographic equations, (1) and (2) could be used to
retrieve horizontal structure in an arbitrary two dimensional airglow profile, where Bi is a single
photometric measurement, ni is the number of atmospheric grid cells sampled by the
measurement, Aij is the path length of the ith measurement through the jth cell, Tj is the retrieved
volume emission rate at grid cell j, and Pij is the expected contribution of volume emission rate Tj
to measurement Bi. In practice McDade and Llewellyn [7, 8] represented the volume emission
rate with Vj, the photometric measurement with Oi and the geometric path length with Lij.
If equation (1) is substituted into (2), with the McDade and Llewellyn [8] variable representation,
then the modified result is
V j( n ) 


 Oi  ni  LijV j( n 1)

 1
i 
( n 1)
  LijV j



j
L
,
(3)
ij
i
where the superscripts (n) and (n-1) indicate the volume emission rate at the current and previous
iterations respectively.
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Submitted to applied optics, June 2002
While equation (3) is complete and consistent with the McDade, Lloyd and Llewellyn [6, 7, 8,
10] formulations it may be simplified with certain assumptions. If Oi, the radiance measurement,
is large compared to ni, the number of volume emission rate elements sampled by measurement i,
then ni may be omitted from the equation. Similarly the -1 term can be eliminated if it is small
compared with the other term in the numerator. If these terms are both removed from equation
(3), then the Vj(n-1) term in the numerator of the summation may be taken outside the summation
as it does not depend on the observations i. The resulting formulation is given by equation (4),
V j( n )  V j( n 1)


 Oi Lij 
i  L V ( n1) 
  ij j

 j

L
.
(4)
ij
i
The denominator in equation (4) is only dependent on the volume emission rate element j and
those observations that sample it. It is the sum of the geometric path lengths of all observations i
that pass through the volume emission rate element j. This summation can be represented by the
value Wj and, as it has no dependence on the observation i, combined with the summation in the
numerator of equation (4) as shown in equation (5).
V j( n )




L
O
ij
i
 V j( n 1)  

( n 1)
Wj 
i   LijV j
 j

(5)
The right hand fractional term contained within the summation in equation (5) divides the path
length, for each observation i through element j, by the sum of the path lengths that intersect that
element. Obviously, the sum of this ratio over all observations i that intersect cell j must equal
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Submitted to applied optics, June 2002
unity. This implies that the term Lij / Wj, within the summation in equation (5), is simply a
normalized weighting factor. For convenience this geometry dependent term is defined as ij.
In their work McDade and Llewellyn [8] used the summation in the denominator of equation (5)
to represent a single line of sight modeled observation based on the previous estimate of the
volume emission rate elements. Although equation (5) contains this single line of sight
representation of a modeled observation any relevant model, Oi(estn1) , can be used. A more
accurate representation of Oi(estn1) would necessarily include the blurring that results from a finite
field-of-view and the spacecraft motion. The final simplified iterative expression is given by
equation (6),
 O

V j( n )  V j( n1)   ( ni1) ij  . (6)

i  Oiest
This equation simply states that the next estimate of the volume emission rate is the current value
multiplied by a weighted average of all observations that sample that cell, divided by their
estimates from the previous volume emission rate solution.
Figure 1 illustrates how the iterative equations manipulate the observations to produce the final
volume emission rate profile. The current iterative estimate, the (n-1)th, of the volume emission
rate profile is used, together with ancillary data (the instrument field-of-view and the satellite
position and pointing), as input to a model that provides the next estimate of the observations.
The estimates of the observations and the actual observations are compared and, as shown in
equation (6), the observations i that sample an element j are divided by their modeled
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counterparts. If the weighted average of these ratios is high (i.e. greater than unity) then the
volume emission rate value contained within that element is raised (i.e. multiplied) by the
weighted average. Similarly if the weighted average is low the contribution from the cell is
decreased. This process is repeated at every cell location to yield a new set of volume emission
rate elements.
Figure 1: A block and data flow diagram for the satellite tomography algorithm.
One technique, or modification of this method, that has been used extensively in Positron
Emission Tomography (PET) medical imaging is the Maximum Likelihood Expectation
Maximization (MLEM) technique [11, 12, 13, 14, 15, 16]. This technique is used to estimate the
vector x (indexed by j) that is a solution to the discrete linear equation Ax = b when the
measurements b (indexed by i) contain noise. The matrix A, with elements aij, is geometry
dependent. The problem may be restated as ‘solve for x such that b    Ax  b   ’. The
common technique used [17] is to maximize the log likelihood function

f x k , x k 1

 
k
  bi a ij x j
   
log a ij x kj 1
k
a
x
i  j   ij  j 
  j

7



k 1 
   a ij x j  (7)
 j




Submitted to applied optics, June 2002
which it has been shown converges, after iteration, to the maximum likelihood solution [18]. The
term x kj denotes the current iterative estimate of x j and x kj 1 indicates the next iterative value,
the summations over i, j and j  are over all possible values. The maximization is achieved by
setting the first partial derivative with respect to x kj 1 equal to zero and then solving for x kj 1 to
give
x
k 1
j


bi
 x 
k
i   a ij  x j 
 j
k
j





a ij 
bi
k

  x j 
.
ij
k
i   a ij  x j 
i a ij 


 j

(8)
Each term in the sum can be interpreted as a weighting filter function term with respect to the
observation i. The MLEM technique is a Multiplicative Algebraic Reconstruction Technique
(MART) where the estimate of x is adjusted through the comparison of the measurements and
the iterative estimate of the measurements. A direct comparison of equations (6) and (8)
indicates that the modified McDade and Llewellyn [8] technique is actually the MLEM
technique for the assumptions defined above.
The Atmospheric Model
The purpose of the atmospheric model is to simulate the observations, shown in Figure 2, that are
made by the measuring instrument. This figure, which is drawn to scale, depicts a satellite in
orbit at 600 km with an optical instrument that simultaneously measures the first 100 km of the
atmosphere along a number of different tangent altitude lines-of sight. This corresponds to a
field-of-view of approximately 2. Although it is difficult to see in the figure the Earth is
considered an oblate spheroid with semi-major and semi-minor axes of 6378 and 6354 km
respectively. This fact is very important for the correct model.
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Submitted to applied optics, June 2002
It is easily seen from Figure 2 that the satellite motion allows regions of the atmosphere to be
sampled from different points in space and, therefore, the line integral measurements for each
observing point follow different paths. A discrete two dimensional grid model of the atmospheric
volume emission rate profile, a discrete model of the satellite attitude and a discrete model of the
optical instrument are all required to represent accurately the process shown in Figure 2.
Figure 2: The geometry of the model drawn to scale. The circles on the outer shell indicate the satellite, the
lines coming from the satellite represent the complete field-of-view of the optical instrument, the dark ring
represents the lowest 100 km of atmosphere and the shaded region represents the solid earth.
The two dimensional grid shown in Figure 3 has two purposes with respect to the tomographic
simulations. This geometric structure, with its angular and radial shell delimited grid cells, is
used to contain an atmospheric volume emission rate profile for the simulated observations and
to store the retrieved profile from the observations. The first situation is referred to as ‘the
observation grid’ while the latter is called ‘the retrieval grid’. Of course the distance, in both
angle and radius, between successive grid cell boundaries is different for each purpose of the
grid. For the simulation results presented here, the dimensions of the observation grid are one
tenth those for the recovery grid.
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Figure 3: The two dimensional volume emission rate grid geometry in a coordinate system that is defined by
the position of the ascending node. A sample line of sight, P asc , is shown. This figure is not drawn to scale.
The grid cells shown in Figure 3 are labeled and indexed by j, this maintains consistency with the
notation introduced in the previous section. In this figure a geocentric coordinate system, in the
orbit plane, has been attached to the grid. This system is called the ascending node coordinate
system because its zero angle is coincident with an arbitrary satellite ascending node crossing.
The satellite position vector S asc and a sample look direction P asc are also shown in this figure.
This coordinate system and its associated vectors are used to compute the geometric path lengths
through each cell; these are required for the integrations involved in the modeled observations.
An ideal observation includes contributions from an infinite number of look directions that make
up the non-stationary (due to satellite motion), spatially finite, complete instrument field-ofview. The first step in the construction of a model observation requires the determination of the
radiance in the direction of a finite number of lines of sight. The second step requires a
calculation of the weighted average of the brightness values, seen along each line of sight, to
determine the simulated measurement. The number of lines used to simulate a single
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measurement dictates its quality, or correctness. The oblateness of the Earth creates severe
digitization errors if only single lines of sight are used for the simulated observations.
Each observation is comprised of many single lines of sight brightness values. The single line of
sight measurement shown in Figure 3 can be described by the summation in equation (9)
Op
asc
 Lp
asc
In this representation L p
V  Lp
,8 8
asc
,j
asc
V7  L p
,7
asc
V  Lp
,10 10
asc
V  Lp
,13 13
V .
asc
,14 14
(9)
is the geometric path length of the line of sight through cell j.
The complete simulated brightness measurement consists of the sum of the weighted average
blurred contributions over all of the lines of sight that constitute the complete field-of-view of a
single pixel. This is seen in equation (10) where the summation is over k and l that designate the
temporal and stationary blurring respectively


Oi      lk Olk  . (10)
l  k

The factor  lk is a normalization term that may be constant for all l and k, or may vary with l in
order to represent any non-uniform sensitivity across the field-of-view. All of the observations
that have been simulated in the results presented here have used seven satellite positions and
seven lines of sight within the field of view of a single pixel. This means that a weighted average
of forty-nine simple line of sight measurements was used to simulate the observation in a single
pixel. An example of the modeled observations that were generated using the input volume
emission rate profile shown in Figure 4a is shown in Figure 4b. For this set of observations the
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Submitted to applied optics, June 2002
optic axis of the imager (pixel 20) stared at a fixed 40 km tangent altitude. In Figure 4a the angle
along the satellite track is defined from an arbitrary ascending node equator crossing. The
composite images assume a linear imager with 100 pixels; i.e. the instrument makes 100
simultaneous measurements with field-of-view centres fixed at uniformly spaced angles over a
2.03 (approximately 100 km in the limb) field-of-view. It should be noted that the volume
emission rate grid shown in Figure 4a is a rectangular representation of the spherical grid shown
in Figure 3 so that the oblateness of the Earth appears as the low frequency oscillation in the
vertical volume emission rate profile. The observation set started with the satellite at 0 along its
track (the ascending node) and includes 2400 evenly spaced measurements as the satellite moves
along approximately one-quarter of an orbit. In this example the first grid cell that is sampled 14
ahead of the first satellite position while the last grid cell sampled is approximately 35 ahead of
the final satellite position.
(a)
(b)
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Submitted to applied optics, June 2002
Figure 4: A sample two-dimensional input volume emission rate grid (a). The grid covers only a partial orbit
and corresponds to the 2400 images taken every half seconds that are shown in (b).
The observation set is just a transform of the volume emission rate grid. As this transform is very
similar to the Cormack transform [19, 20] it closely resembles a two-dimensional function with
an exact inverse, i.e. an exact two-dimensional solution exists for a geometry that is nearly that
of the satellite measurements.
The two-dimensional volume emission rate grid shown in Figure 4a has the same vertical profile,
with respect to altitude above the surface of the earth, at each angle as its basis. However, we
have added some angular and vertical structure in order to test the inversion capabilities of the
tomographic algorithm. This superimposed structure has been chosen in order to simulate
potential wave like structures. The form of the multiplicative structure is given by equations (11)
and (12),

e  r rmin  
e
M  1   Amin 
H 

  0 2
2 2
Amax  Amin 
cosk r r  cosk  , (11)
e H
.
H
(12)
The wavenumbers k r and k  in equation (11) represent the radial and angular oscillations
respectively. Throughout the present work the wavelength in the radial dimension was fixed at
10 km while k  was allowed to vary so that the angular wavelength ranged from 0.5 to 3.0.
These angular oscillations correspond to horizontal structures with scale sizes between 50 and
300 km. The other two terms in equation (11) are an exponential that represents a Gaussian
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decay from some central angle,  0 , and a growth term with respect to radial distance from the
center of the Earth. This latter is represented by the formulation in braces.
For test purposes  0 was set to the angular centre of the grid and  was chosen to give a decay
half-width equal to 20. The radial growth term is slightly more complicated. H is the total radial
extent of the grid, rmax – rmin, and the growth term  in equation (11) is determined from equation
(12). For the present work Amin and Amax have been defined as 0.2 and 0.8 respectively. The
growth term results in oscillations that increase with radial distance from Amin (at the bottom of
the grid) to Amax (at the top of the grid). This structure is illustrated in Figure 5, which shows
two cross-sectional views from Figure 4a.
Figure 5: The structured and unstructured profiles compared in cross-section.
One objective of the present work is to determine the minimum horizontal scale size of structure
that can be resolved with the tomographic technique described in the previous section. To this
end the structure resolution simulations have been made for different imaging rates and for
different measurement noise levels.
Simulation Results
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One important consideration in the tomographic retrieval process (equation 6) is the number of
times that each grid cell is sampled. This is closely related to the imaging rate so that each of the
noise and scale size simulations was made for 8 different imaging rates. These are 0.5, 0.66,
0.75, 1, 1.5, 2, 2.5, and 3 seconds between images and correspond to 2400, 1800, 1600, 1200,
800, 600, 480 and 400 images respectively as the satellite travels approximately 75 along its
track. This is the distance that the satellite at 600 km altitude travels in 1200 seconds.
An example of the tomographic retrieval for 3400 images is shown in Figure 6. The observation
set and the corresponding two-dimensional volume emission rate profile are those shown in
Figure 4a. The small scale wavelike structure in Figure 4a corresponds to equation (11) with an
angular wavelength of 1. The plots shown in Figures 4a and 6 should be identical if the
inversion is perfect. While it is clear from Figure 6 that the small scale wavelike structure can be
identified it is not obvious that these perturbations are fully resolved. The inaccurate retrieval at
each edge of the recovery grid is also evident in Figure 6. These poorly retrieved ‘edge-effect’
values extend for approximately 22 from either edge of the grid and are the result of undersampled grid cells in these regions. Both the vertical and radial cross-sections of the input and
the retrieval are shown in Figure 7. It is clear that the structure is identified, although the
amplitude of the structure is damped.
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Submitted to applied optics, June 2002
Figure 6: An example of the retrieved volume emission rate profiles that are analyzed throughout this work.
The retrieval indicates the desired structure in the interior of the grid while at the angular extremes the
retrieval is inaccurate due to under-sampling of the grid cells. The input grid and the observations are those
shown in Figure 4.
Figure 7: Cross-sections taken from the input and retrieved grids shown in Figures 4b and 6 respectively. The
structure is well identified and the edge effect is clearly visible in the radial cross section plot.
Noise Considerations
The tomographic process has been outlined in both Figure 1 and equation (6). This equation
states that all observations that intersect a grid cell contribute to the iterative value of the volume
emission rate in that cell. The observations contribute in a fashion that is consistent with a
weighted average taken over the ratio of the actual to the modeled measurements. Therefore, it
can be expected that the larger the number of observations that contribute to the weighted
average then the smaller will be the effects of random errors due to measurement noise. This
hypothesis has been tested with the eight different imaging rates and six different relative
Gaussian random noise conditions.
The modeled observations were generated using a two-dimensional volume emission profile that
did not include any superimposed structure [equation (11)]. Eight sets of high resolution,
noiseless observations, one for each imaging rate, were generated for the angular range from 14
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Submitted to applied optics, June 2002
to 109 in front of the initial satellite position. Five different noise levels, Gaussian random with
standard deviations calculated from the signal to noise ratios, were added to each of the eight
measurement sets. In this way forty observation sets were available for simulated retrievals. The
Signal to Noise levels were set to 100:1, 50:1, 20:1, 10:1 and 5:1.
The benchmark used to evaluate the quality of the retrieval process was the standard deviation
associated with a percent difference histogram. This is the histogram derived from the calculated
percent difference between the input and retrieved volume emission rates for each of the cells in
the angular interior of the grid, i.e. away from the edge effects. Examples of these histograms,
for two different imaging rates and noise levels, are shown in Figure 8. It is readily apparent that
the higher imaging rate histogram, Figure 8a, is much narrower than for the lower imaging rate
histogram, Figure 8b. This is mainly due to the noise levels associated with the input
observations, although there is a small contribution from the imaging rate (Figure 9). Both
histograms shown in Figure 8 are approximately Gaussian and centred on 0%. Any deviations
from pure Gaussian histograms, centred on zero, are due to systematic errors that are related to
interpolation from the high resolution observation grid to the low resolution retrieval grid. These
grids use cell sizes of 0.02 by 100 m and 0.2 by 1000 m respectively.
(a)
(b)
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Submitted to applied optics, June 2002
Figure 8: Histograms of the percent difference calculations for two different retrievals. Both histograms are
approximately Gaussian and centred near 0%. The histogram (b) that corresponds to the lower imaging rate
and higher noise level is considerably wider.
The standard deviations of the percent error histograms for the retrievals done with 32 of the
observation sets are illustrated in Figure 9; the error histogram standard deviations are given for
the eight imaging rates. It is apparent that for each noise level the error rate decreases with an
increase in imaging rate. The decrease in the error rate is marginal for the 1% noise level but is
quite significant for noise levels of 5% and higher. Thus faster imaging rates imply better noise
reduction with the tomographic technique.
Figure 9: The standard deviations of the percent error histograms for 32 retrievals done with different
imaging rates and various noise levels. This figure illustrates the noise reduction capabilities of the
tomographic algorithm. The standard deviation decreases, independent of the noise level, as the imaging rate
increases.
Structure Considerations
The primary reason to use the two-dimensional tomographic technique to retrieve atmospheric
volume emission rate profiles is to identify structure in both the vertical and angular (along
track) dimensions in the satellite orbit plane. Therefore, it is of value to determine the lower limit
of the scale size of wavelike structures, defined according to equation (11), that can be fully
resolved with this technique.
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The noiseless input observation sets used for this study included data from each of the eight
imaging rates and four structure scale sizes. The simulated measurements in a single pixel, for
each of 400 images, are shown in Figure 10. Each of the panels in this figure illustrates a
different structure scale size. It is clear that as the scale size becomes smaller so the perturbation
on the measurements also decreases even though the amplitude of the input perturbation is
constant. This is due to the removal of the high frequencies by the line of sight integration. The
perturbations for the 0.5 structure, when plotted on the same scale as those for the 3 structure,
are hardly visible in the presence of the background signal. This is of course more of an issue
when observational noise is considered.
Figure 10: Pixel cross sections or measurements from a single tangent altitude. These measurements indicate
that the perturbations due to the structure imposed on the input volume emission rate profiles become more
blurred through the line of sight integration as the scale size of that structure decreases.
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The comparison of six constant altitude cross-sections from separate retrievals with their
corresponding input profiles is shown in Figure 11. Again these comparisons are for the inner
parts of the grid where the structure has been imposed. The cross-sections are offset from each
other in order to illustrate the retrievals. The four separate data sets shown in Figure 11a
correspond, from top to bottom, to 0.5, 1, 2 and 3 (50, 100, 200 and 300 km) wavelike
structures for the highest imaging rate. The retrievals for observations made at the lowest
imaging rate, with the identical volume emission rate profiles, are shown in Figure 11b. As noted
the simulated observations excluded random measurement noise and for this condition there are
no obvious differences in the quality of the retrievals for the different imaging rates.
(a)
(b)
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Figure 11: Radial shell cross section comparisons of the input and retrieved volume emission rates for
angular structure of 0.5, 1, 2 and 3. The 1 and larger structures are fully resolved. These simulations are
for the highest imaging rate, 2400 images (a), and the lowest imaging rate, 400 images (b).
It is obvious from the plots shown in Figure 11 that the small scale 1 structure is identified and
resolved by the tomographic technique, although the 0.5 structure is not identified. However, it
is also clear from a comparison of these plots that the identification of small scale structure with
the tomographic retrieval technique is not significantly altered by the imaging rate, in the
absence of measurement noise.
However, as any real instrument system necessarily includes noise the same retrievals were
attempted in the presence of noise. One set of simulated measurements is shown in Figure 12.
These are the same as those in Figure 10 but modified with a two percent Gaussian random
measurement noise component. It is readily apparent that the perturbations that were present in
the absence of observational noise have been obscured. This is especially true for the smaller
scale structures.
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Figure 12: Pixel cross-sections, or measurements, from a single tangent altitude that contain two percent
Gaussian random noise. The perturbations due to the structure imposed on the input volume emission rate
profiles are barely visible even for the larger scale structures.
The results for the corresponding tomographic retrievals are illustrated in Figure 13. The upper
plot, Figure 13a, corresponds to retrievals made with measurements simulated at the highest
imaging rate while the lower plot is for modeled observation at the lowest imaging rate. All of
the shown comparisons, of the input and retrieved profiles, are for the 1 wavelike structure.
Larger scale sizes retrieve as well, or better, while smaller scale sizes are not resolved even in the
absence of noise (Figure 11). Higher noise levels introduce systematic, as well as random, errors
in the retrievals as any simulated measurements that drop below zero are not included in the
retrievals and, therefore, introduce a positive bias in the measurements. It is apparent that the
noise reduction properties of the two-dimensional technique are better for the higher imaging
rate.
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(a)
(b)
Figure 13: Radial shell cross-section comparisons of the input and retrieved volume emission rates for
angular structure of 1 and four different noise rates. These simulations are for the highest imaging rate,
2400 images (a), and the lowest imaging rate, 400 images (b). The signal to noise in the measurements for each
of the two plots increases, from top to bottom, from 5:1 to 50:1
When the imaging rate is the highest the match between the input and retrieved profiles for the
1 structure is quite good even for 10% noise levels (10:1 S/N, second line in each panel). The
quality of the retrieval, at this S/N ratio, is visibly degraded for the lower imaging rate. It is of
value to note that even for the 20% noise level and the highest imaging rate the 1 structure
information is not lost in the retrieval. This is further illustrated in Figures 14a and 14b where a
simple two-dimensional, three-element wide, boxcar filter has been applied to the retrieved twodimensional profiles that contain the 3 structure and the same observational noise levels as in
Figure 13. The structures are clearly identifiable in all cases and the retrievals are good even at
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the 20% noise level. This implies that the two-dimensional tomographic technique tends to
localize the random errors in the retrieval grid and not disperse adverse effects throughout the
recovery space.
(a)
(b)
Figure 14: Radial shell cross-section comparisons of the input and retrieved volume emission rates for
angular structure of 3 and four different noise rates. The raw retrievals in (a) have been smoothed (b) with a
very simple three-element wide boxcar filter. These simulations are for the highest imaging rate of 2400
images.
It is clear that the two-dimensional tomographic retrieval technique is effective for the
identification and resolution of small scale structure even in the presence of substantial random
measurement error. For the assumed instrument parameters the lower limit for fully resolved
wavelike structure appears to be about 1. However, scale sizes smaller than this limit may be
identifiable, with a modified instrument, as the simulations do not place a lower limit on the
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scale size structure that can be resolved with this technique. It has only been demonstrated that
for the assumed experimental setup, and the structure represented by equation (11), that the
lower limit on the scale size that is fully resolvable is 1.
Conclusion
In the present work we have shown that the tomographic technique for satellite volume emission
rate tomography that was developed by McDade and co-workers [6, 7, 8] is closely related to the
Maximum Likelihood Expectation Maximization technique that is used in medical imaging. It
has been demonstrated that the tomographic technique is well suited for the retrieval of structure
in two-dimensional airglow profiles contained within the orbit plane of a satellite in a 600 km
circular orbit. The adopted algorithm is able to resolve fully 10 km vertical structure and identify
horizontal wavelike structure as small as 1 from 100 simultaneous different tangent altitude
measurements collected at each satellite location. It has also been shown that in the presence of
observational noise faster image rates improve the quality of the retrieval. However, in the
absence of any random noise the imaging rate does not significantly affect the ability of the
retrieval technique to resolve small scale size structures. It is intended to apply this tomographic
technique to the retrieval of atmospheric structure from observations made with the InfraRed
Imager part of the OSIRIS instrument on the Odin satellite.
Acknowledgement. This work has been supported by the Natural Sciences and Engineering
Research Council of Canada and by the Canadian Space Agency.
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