Heliospheric tomography using interplanetary scintillation
observations
B.V. Jackson, P.L. Hick
Center for Astrophysics and Space Sciences, University of California,
LaJolla, California
M. Kojima and A. Yokobe
STElab, Nagoya University, Toyokawa, Aichi 442, Japan
Abstract.
We have produced a Computer Assisted Tomography
(CAT) program that optimizes a three dimensional solar wind velocity,
V, and density fluctuation, Ne, model to fit observed
interplanetary scintillation (IPS) data from Nagoya, Japan and Cambridge,
England. The multiple perspective views of the solar wind needed
for the reconstruction solution are provided by solar rotation
and outward solar wind motion. The CAT program iterates to a least
squares fit solution to the observed IPS values. We map the model
to one AU and compare this to in situ observations from
the IMP spacecraft. From this comparison we find Ne
is proportional to Ne0.3. We plot
the optimized model as Carrington maps in velocity and Ne,
and compare these with Yohkoh Carrington synoptic maps. We find
that the model velocity projected to the solar surface for individual
rotations shows regions of high velocity that map directly to
coronal hole areas observed in Yohkoh Soft X-ray Telescope (SXT)
observations. Regions of slow velocity generally map to bright
regions in SXT data. Regions of high Ne show
a high correlation with regions of high solar activity observed
as bright in Yohkoh SXT observations.
Since the 1960's interplanetary scintillation (IPS) measurements
have been used to probe solar wind features with ground-based
meter-wavelength radio observations (Hewish et al., 1964; Houminer,
1971). These observations from the University of California, San Diego,
(UCSD) (Coles and Kaufman, 1978)
and Nagoya (Kojima and Kakinuma, 1987) multi-site scintillation
array systems have been used to determine velocities in the interplanetary
medium since the early 1970's. The scintillation-level intensity
IPS observations, which arise from small-scale (200km) density
variations, highlight heliospheric disturbances of larger scale
that vary from one day to the next and are often associated with
geomagnetic storms on Earth (Gapper et al., 1982). These scintillation
level IPS observations show a predominance of disturbances that
appear to co-rotate with the Sun as inferred from a list of events
and their associations (Hewish and Bravo, 1986).
Carrington maps of velocity and scintillation level are one of
the techniques used to display these interplanetary observations.
Until now, these maps have been produced using the so-called point-P
assumption, where the material responsible for the observed phenomenon
is assumed to be located at the point where each line of sight
passes closest to the Sun (i.e., Coles et al., 1980; Rickett and
Coles, 1991; Hick et al., 1995). In this approximation each observation
is related to a single solar latitude and longitude. By combining
many observations it is then possible to determine the solar origin
of heliospheric structure in an approximate fashion. For instance,
from IPS velocity data it was determined that the solar wind in
polar coronal holes has high speed (Kakinuma, 1977; Coles et al.,
1980, Kojima and Kakinuma, 1990) long before observations from
the Ulysses spacecraft (McComas et al., 1995) measured these velocities
in situ. Regions of slow solar wind are generally found
near the solar equator especially at solar minimum, and thus near
the location of the magnetic neutral line as determined by the
potential magnetic field model (Hoeksema et al., 1983). Scintillation
level data from the Cambridge scintillation array have been analyzed
in the same manner (Hick et al., 1995). The detailed maps produced
by this technique show that the solar polar regions generally
do not scintillate very strongly compared to regions near the
solar equator. Houminer and Gallagher (1993) find that some of
the regions which scintillate strongly and corotate or return
from one rotation to the next are located near the heliospheric
current sheet. Hick et al. (1995) have determined that solar active
regions generally bright in X-rays, and not the current sheet,
are the solar surface locations of corotating regions observed
to scintillate strongly in the Cambridge IPS observations.
Tomography methods can be used to circumvent the rather crude
point-P approximation. IPS observations covering a large range
of solar elongations and obtained over an extended period of time
provide a global view of the inner heliosphere. The multiple perspectives
required for tomographic analysis are provided by both solar rotation
and outward solar wind motion. Detrimental effects from evolution
of solar structures as they rotate during the period of observation
can be minimized by selecting observations from a quiet part of
the solar cycle. Using scintillation theory to relate IPS observations
to the distribution of density variations along the line of sight,
iterative tomographic methods can be used to find the three dimensional
heliospheric model of density variations, producing line-of-sight
integrated model IPS observations matching the actual IPS observations
as nearly as possible.
Tomography is best-known for its application in the medical profession,
where it used as a non-invasive way to probe the human body, and
reconstruct its internal structure in three dimensions (Gilbert,
1972). One of the earliest uses of tomography was in solar radio
astronomy (Bracewell, 1956). Other areas where tomographic reconstruction
techniques have been successfully applied are in studies of binary
star systems (Marsh and Horne, 1988) and accretion disks in astrophysics
(Gies et al., 1994), acoustic sounding in oceanography (Worcester
et al., 1991), seismic studies in geology (Anderson and Dziewonski,
1984), auroral studies (Frey et al., 1996) and coronal studies
in solar physics (Hurlburt et al., 1994). An application in atmospheric
modeling, somewhat similar to our model in its use of an irregular
sampling of refractometric sounding observations is discussed
in Gorbunov (1996). In general, depending on the object resolution
and noise in the data set, when more perspective views of an object
are available, a finer resolution of its three dimensional structure
becomes possible. In medical applications it is generally possible
to obtain many views from as many different directions as deemed
necessary. Most other tomographic applications are limited in
the ability to view objects from a large number of directions.
However, in theory even a single perspective view can give information
of an object's three dimensional extent (Katz, 1978).
In coronal and heliospheric physics there have been numerous attempts
to reconstruct the corona and heliosphere in three dimensions.
Near the Sun there is good reason to determine the three dimensional
shapes of structures. Coronal mass ejections (CMEs) often have
a loop-like appearance. As helical loops driven by currents as
proposed by Anzer (1978) or Mouschovias and Poland (1978) the
shape of a CME should follow a very specific pattern. If, however,
a CME is a spherical bubble, then it might very well be the remnant
of a large addition of energy at a single point in the low corona
(Wu et al., 1976). Various techniques used to determine CME shapes
from the single perspective of Earth, including polarization (Munro,
1977; Crifo et al., 1983) and depletion of the corona (MacQueen,
1993) indicate that CMEs are extensive coronal structures. Studies
using multiple perspectives from spacecraft viewing from different
vantage points (Jackson et al., 1985) reach the same conclusion.
A CAT analysis of CMEs by Jackson and Hick, 1994) and Jackson
and Froehling, 1995) performed on Solwind coronagraph and Helios
spacecraft photometer observations taken from widely different
perspectives show the extended three dimensional shape of CMEs.
The shapes and positions of coronal streamers can indicate their
location and extent relative to the magnetic structures on the
Sun, and this in turn can give an indication of whether all streamers
are formed by the effects of a global solar current `pinch' effect
or some more local magnetic phenomena. Some of the first coronal
tomographic analyses from Skylab coronagraph observations (Wilson,
1977; Jackson, 1977) used solar rotation to provide perspective
views of the corona. These analyses have been enhanced recently
by Zidowitz et al. (1995) using similar techniques to reconstruct
coronal densities from Mark III coronagraph observations. Previous
heliospheric tomography from IPS has been attempted without the
aid of sophisticated computer techniques. These analyses (Gapper
et al., 1982; Behannon et al., 1991) have relied on a combination
of solar rotation and outward motion to determine the three dimensional
extent of coronal structures. In these analyses, models of different
three dimensional coronal structures were used to determine their
appearance on a two-dimensional image. The model shape chosen
to best represent the data was that which best fit the image by
eye.
In the analysis presented here, the tomography performed is most
like this technique. However, in this analysis we iterate to solve
a three dimensional model using a least squares iterative technique
to fit each IPS observation. The resolution of the three dimensional
structure is limited by the amount of data and its inherent noise.
Because of these limits, we present our analysis in the form of
Carrington synoptic maps projected to one height (to 1 AU, or
to the solar surface).
In section 2 of this paper, we present the analysis techniques
we have used to deconvolve the IPS data and the assumptions required
to apply these techniques. In section 3 we describe the computer
program used to perform the analysis and how it is used to fit
the observed data. As described in section 4, the analysis is
used to extrapolate the information to 1 AU to compare with IMP
spacecraft data at Earth. In section 5, the deconvolved model
is shown in the form of Carrington synoptic plots, and these are
compared with Yohkoh X-ray and Sacramento Peak Fe XIV intensity
maps at the solar surface. We conclude in section 6.
The IPS technique relies on a number of assumptions to map variations
of intensity and velocity along each line of sight. In weak scattering
(assumed here exclusively) the Born approximation holds, and the
diffraction pattern can be regarded as a sum of contributions
from each thin scattering layer perpendicular to the line of sight
(Tatarski, 1961). Tomography likewise relies on a number of assumptions
which are needed for the technique to work. In the following subsections,
we present the assumptions and ideas we have used to analyze the
IPS data.
The Cambridge disturbance factor, G, is available from
several hundreds of sources measured each day for the period March
1990 to September 1994. For the analysis presented here, we have
used data from a relatively short time interval during the summer
and fall of 1994. The data were selected from this period because
solar surface features from consecutive rotations show little
evolution from one rotation to the next (as might be expected
in the declining phase of the solar cycle). Carrington maps produced
from consecutive solar rotations using IPS observations also show
a great deal of similarity from one solar rotation to the next
during this period.
The value of G is defined as,
(1)
where m is the scintillation level and <m>
is the mean level for the observed source at that elongation.
Scintillation level measurements from Cambridge are available
as G-level measurements at a given time and sky location.
For each day we plot these levels as an image of the sky and determine
whether any obvious noise is present in the data. In an editing
procedure developed at UCSD, we are able to remove obviously noisy
data. These locations most frequently involve regions near the
solar hour angle which are presumably solar radio noise. We also
find that regions from sunset to sunrise (centered on midnight)
sometimes have high G-levels. These are assumed to be regions
contaminated by ionospheric scintillation, and if this occurs
we eliminate the daytime region of the map adjacent to the nighttime
portion. For the measurements for this period of the solar cycle,
fewer than 1 % of the observations were removed from the record
by our editing procedure. In practice, experience with these analyses
has shown that we are able to obtain consistent results by including
sources from Cambridge observations from 30 to 80 elongation.
In our analysis we presume a measurement at one elongation is
as valid as at any other, and weight each equally.
The strength of the scintillation level along the line of sight
from Earth depends upon several different factors. Near the Earth,
the closeness of the phase screens to the Earth reduces the level
of scintillation. This fresnel filtering effect (Coles and Harmon,
1978) provides a cut-off in the line of sight scintillation response
near Earth. At distances of several AU from Earth, the source
size at radio frequencies decreases the IPS response along the
line of sight. Sources of larger angular size scintillate less
at the same distance from Earth than do those of small angular
size for any given radio frequency. This weighting factor along
the line of sight WC(z) can be approximated
in weak scattering (Young, 1971) by,
(2).
The terms in this integral are q, the wave vector (here
given the typical average power index of 3), and 0
= 0.3 arc-second, the average angular size of a radio source at
the wavelength =3.68 m (81.5 MHz) of the Cambridge IPS observations.
The solution of the integral for WC(z) is plotted
for z from 0.0-2.0 AU (from Earth) in Figure 1. The scintillation
level m is related to the density variation along the line
of sight by,
(3).
Here, Ne(z) is the electron density variation
at distance z along the line of sight. The density variation
values along the line of sight are not known beforehand, but we
assume that the density variation scales with a power law of heliospheric
density,
(4).
The power P is varied to best fit observations. For the purpose of the tomographic program, we assume that Eq. (3) can be digitized as,
(5),
where our step-size z is chosen to be 0.05 AU. We extend
our line of sight for each source to a position of 2 AU from Earth
(n=40) at any elongation. This digitization of the line of sight
is sufficient to match a map resolution of 10 at the smallest
elongations considered.
Fig. 1. Line of sight weighting factors for Cambridge and
Nagoya IPS.
Valid Nagoya IPS velocity data are available from approximately
15-20 radio sources each day. The data are available, edited by
the group at Nagoya. In the tomography program, we assume that
extremely high velocities are not valid and discard those measurements
that are higher than some limit (820 km/s is commonly used) from
the analysis. In general, a limit of 820 km/s removes fewer than
0.5 % of the radio sources from the record for Carrington rotations
1884 - 1886. We include sources at all elongations.
The IPS velocity cross-correlation follows a similar relationship
to that of the intensity scintillation. We approximate the velocity
observed at Earth as,
(6)
where V(z) is the component of the solar wind velocity
perpendicular to the line of sight. The quantity Ne(z)
is the electron density variation at distance z along the
line of sight as determined by the intensity scintillation from
the Cambridge array over the same time period. WN(z)
is the weighting factor determined as in Eq. (2) using a value
of 0 = 0.1 arcsecond, the average angular size of a
radio source at the wavelength =0.917 m (327 MHz) of the Nagoya
IPS array observations. The solution of the integral for WN(z)
is plotted for z from 0.02.0 AU (from Earth) in Fig. 1.
Equation (6) digitizes as before to,
(7)
with step-size and distances in the analysis chosen the same as
for the Cambridge IPS analysis.
The tomography program applies corrections to an existing model,
modifying the model until there is a least squares best fit match
with the observations. The assumptions which make up the model
are therefore some of the most important parts of the analysis.
The solar wind model considered provides two parameters to fit
in three dimensions: the radial outward solar wind velocity and
the solar wind small-scale density variation values which produce
the intensity scintillation.
The density variation values are related to one another over different
heights assuming the velocity from one height to another is known
and that the relationship of the density variation to density
with height is known at all heights. In the model we use, the
density variation is assumed proportional to some power of the
density (Eq. 4). In the current analysis, this relationship is
not allowed to vary with height. The density changes with height
as velocity and solar wind expansion dictate. If the velocity
is assumed everywhere constant within the model, the density would
change as r-2 with distance r from the
Sun.
The velocity from the scintillation observations is mapped from
a reference height in the model by projecting it radially outward
(and inward) from this height. The solar wind motion is assumed
to be strictly radial, and thus, for example, faster solar wind
catches up with slower wind that is flowing along its Archimedean
spiral. The velocity is averaged, and then projected outward at
this new velocity. The different velocity implies an accumulation
of mass, and continuity of mass is evoked within a latitude band
to modify (increase) the density accordingly. The opposite occurs
for locations where the slower velocities trail. In this instance
the velocities are averaged and the rarefaction within the latitude
band implies a mass depletion from r-2 expansion.
At each height the velocity structure of the Carrington map is
filtered so that some information from neighboring latitudes and
longitudes is retained. The longitudinal filtering is increased
in the model linearly with the cosecant of the latitude since
the resolution of the model in the spherical coordinate system
used effectively increases spatially with increasing latitude.
Consistent with in situ spacecraft observations (Hundhausen
et al. 1970), radial outward motion of the plasma is assumed in
the model. In addition, the model is fit assuming the structures
observed do not change except for outward motion over the time
they are observed - i.e., that solar corotation is valid over
the time interval of the observing period. This interval is typically
half a solar rotation for any given heliospheric coordinate position
and is dictated by the amount and quality of the available observations
and the map resolution. For one solar rotation there are typically
5000 G-level lines of sight. In theory this implies a possibility
of determining the density variations for 5000 model locations.
The velocity measurements typically number fewer than 600 for
each rotation with a consequent fewer number of potential model
locations for which to determine velocities. The computational
aspects of the tomography are discussed further in the following
section.
The computational aspects of the tomographic program necessarily
deal with the detailed geometry for each line of sight: the location
of each within the three dimensional solar wind model and its
projection to a specific height. A line of sight projects to a
given height as shown for several examples in Fig. 2. In the point-P
approximation, a single position along this line is mapped to
a given height. The observed IPS value is placed at that coordinate
position (in a Carrington map) formed at that height. If more
than a single observation falls within the resolution element
of the mapped coordinates, it is usually averaged with a weight
of one for each observation. To construct observations from a
point-P map, one would trace P for the given line of sight to
the map and give the line of sight the appropriate model value.
Fig. 2. a) Schematic of a line of sight and its projection
to a constant height from the Sun. The dashed line on the reference
surface is the line of sight projection after taking outward solar
wind motion into account. b) Lines of sight projected to this
same height in Carrington coordinates. Three lines of sight are
depicted. Note the different perspectives indicated by each as
locations where the lines of sight cross.
In the tomographic analysis scheme used here, the Carrington map
at a given height provides an input to the line of sight at each
point along it. Each of the digitized points is weighted according
to the scheme presented in the preceding section. These weighted
values derived from the Carrington map constitute the model values
for each observation. Both a model G-level and V
are calculated in this way. These model values are then compared
with the observed values for each source measured, and a ratio
formed. At this point a difference of the ratio from one is accumulated
for each source. If the model agrees with the observation exactly,
the ratio is one. If the observed values are greater than the
model values, this indicates in this instance that each model
value along that line of sight should be increased in order to
fit the observations.
For each point in the model (in this case on the Carrington map
at the given height), the changes from model values are accumulated,
weighted by the scheme which was used to produce the model values
in the first place. The weights are also accumulated. After all
the observed to model source ratios are formed, the map model
is changed in a least-squares sense according to the accumulated
weighted corrections for each. These least squares differences
from the mean for each coordinate location are summed and used,
along with the differences of the source observation to model
ratio differences, to indicate the rate of convergence. In the
tomographic analysis used here, first velocity and then density
variation is changed. At the end of each change of the mapped
model velocities, the three dimensional solar wind model is recalculated.
This is done to assure that the newest values of velocity and
mass from the assumption of mass continuity are used to determine
the proper density variations along the line of sight and an accurate
velocity mapping for use in the determination of more refined
model G-levels. In order that several different perspective
lines of sight produce the modeled values, we require that several
lines of sight contribute to each coordinate position resolution
element on the Carrington map in order that it be changed. In
practice we typically require that 3 or more lines of sight contribute
to changes in the velocity model and 10 or more lines of sight
contribute to changes in the density variation model.
The computer iterates to a solution, generally converging to an
unchanging model within a few iterations. For a typical rotation,
a set of iterations generally takes about ten seconds on a VMS
alpha machine (DEC 3000 Model 400). We typically operate the program
for 9 iterations for good measure, and then remove those few sources
from the data set which do not fit within a three-sigma limit
of the source model to observed ratio mean for both V and
G-level. The program is then allowed to operate for another
9 iterations. Tests of the program show that the model solutions
are not sensitive to the starting input model, and that after
a few iterations any signature of the input model is lost. Other
tests show that map inputs can be reproduced in the tomography.
Notably, density or velocity spikes in the Carrington maps at
a single resolution element location take approximately 9 iterations
to be recovered within 10 % of their original value. Smooth variations
in density or velocity generally take fewer iterations to recover
values to a 10 % level.
We do not know how well the tomographic program reproduces three
dimensional solar wind densities and velocities simply because
there are few observations with which to compare them. Indeed,
the IPS observations are best mapped to heights nearest the location
of the observations - namely at approximately 0.5 AU. Few spacecraft
have ever measured plasma densities at this solar distance, and
none have measured solar plasma parameters inside 1 AU out of
the ecliptic plane. However, the tomographic model densities and
velocities are available in a three dimensional model and can
be extrapolated to any height, presumably with less success as
the extrapolation becomes greater. Furthermore, there is a free
parameter available in the analysis that is not well-known. This
parameter is the power P in the relationship NeNeP.
Others (e.g., Tappin, 1986; Zwick, l988) have estimated this value,
and thus we have some experience from past analyses of what it
should be. From Eq. (1), (2) and (4) in our analysis, we determine
the proportionality,
(8)
Using Cambridge scintillation data from the time period 1978 -
80, Tappin (1986) found this proportionality to have a value of
P=0.5.
Fig. 3.
Rotation 1884 density time series from the IPS model projected to 1 AU compared
to the density time series from the IMP spacecraft (dashed line).
Fig. 4.
Density correlation. The line segment through the data points gives the least
squares best fit. The dashed line is the line of one-to one correspondence.
Fig. 5.
Rotation 1884 velocity time series from the IPS model projected to 1 AU compared
to the velocity time series from the IMP spacecraft.
Fig. 6.
Velocity correlation. The line segment through the data points gives the least
squares best fit. The dashed line is the line of one-to-one correspondence.
By mapping the data measured in our models to 1 AU, we are able
to compare it directly to the density values measured by the IMP
spacecraft near Earth. For this comparison, we average the IMP
data into 18-hour averages, consistent with the spatial resolution
present from the longitudinal binning of the tomography data into
10 averages. The densities mapped to 1 AU are shown as two time
series in Figure 3. These densities were fit by adjusting the
value of P in 0.05 increments so that the excursions of
the two time series fit best by eye for this rotation. We found
a value of P=0.3 fit the observations of rotation 1884 the best.
Note that a large density peak appears in the time series from
the IMP spacecraft at day 177 that is not reproduced in the model
value projected to 1 AU. We suspect that the peak in the IMP data is
produced by a transient heliospheric density enhancement (such as
a CME), which is not reproduced correctly by the tomographic
reconstruction). If one ignores the data for the days
near this time, the correlation coefficient between the two values
of density (Fig. 4) is 0.6. We find that the location and shapes
of the structures mapped in the tomographic analysis are not very
sensitive to this parameter nor is the convergence of the model,
at least within the range of values which fit the observed data.
We expect to further refine this relationship by measuring other
rotations and by using more sophisticated analysis techniques.
Figure 5 gives two velocity time series from Carrington rotation
1884. We see that the temporal sequence from both time series
is fairly similar. A correlation of data points mapped one to
another is shown for velocities in Fig. 6. The correlation coefficient
for this set of velocities is 0.4. Velocities determined by the
mapping should ideally correspond one-to-one with in situ
measurements, and we see that within the measurement errors this
is the case. We find that large variations of the density P
parameter have little effect on the model velocities. We plot
the 1 AU Carrington maps from which these data are interpolated
in Figures 7a and 7b.
Another comparison to be made, and the reason we attempted the
tomography in the first place, are the correlations of densities
and velocities with structures at the solar surface. Figures 7c
and 7d map rotation 1884 in terms of velocity and density where
they are compared with Carrington synoptic maps derived from Yohkoh
Soft X-Ray Telescope (SXT) and Sacramento Peak Fe XIV brightness
measurements (Figures 7e and 7f). To compare density values projected
to the solar surface with surface features, we have removed a
r-2 density component from them. Outward velocity
is not well known within a few solar radii of the Sun. Thus, we
expect a small amount of longitudinal error in this mapping for
both velocity and density variations caused by the unknown amount
of acceleration and rigid rotation of the solar corona near the
solar surface. The solar surface densities are inaccurately located
by a few degrees in longitude because of this. As in the point-P
maps presented by Hick et al. (1995) and Hick and Jackson (1995),
for rotations 1853-57 and 1884-86, the comparisons show great
similarities in the locations of the active regions and regions
that scintillate strongly and have low velocity.
Fig. 7. Carrington map displays for rotation 1884.
(a)
Deconvolved velocity projected to 1AU.
(b)
Deconvolved density projected to 1 AU.
(c)
Deconvolved velocity projected to the solar surface.
(d)
Deconvolved density projected to the solar surface.
(e)
Sacramento Peak Fe XIV brightness map derived from East and West
limb scans. The potential field derived neutral line is superposed on the map.
(f)
Yohkoh Solar X-ray Telescope Carrington map derived from
central meridian solar images and degraded to 10 by 10 resolution,
also including the neutral line.
In comparison with in situ data at the ecliptic, the tomographic
analysis appears to give superior results to the point-P analysis
others (i.e., Houminer and Hewish, 1974; Coles et al., 1978; Jackson
and Rickett, 1988) have presented in the past. In the case of
velocity, the point-P analysis typically underestimated the higher
velocities. This does not appear to be the case in our tomographic
comparisons. The point-P analysis also underestimated polar velocities.
The reasons for this were generally believed to be caused by the
high density variations and generally slower velocities near the
ecliptic plane. Occasionally, when views were available looking
poleward outside of the high levels of scintillation near the
ecliptic, actual polar velocities and low scintillation levels
over the pole could be measured accurately (Jackson et al. 1994).
The tomographic analysis by its very nature handles both the nearby
density variations and those distant from Earth as accurately
as the modeling allows, and least squares fits data to observations
over the whole model.
Carrington maps produced using the tomographic analysis indicate
velocities over the solar poles that map in detail to the coronal
hole regions and their boundaries. These high solar wind speed
regions occasionally map to lower latitudes from the polar regions
across the solar equator. Seldom are the velocities in these maps
greater than 800 km/s, consistent with findings from the Ulysses
spacecraft as it crossed polar latitudes at distances greater
than 1.5 AU from the Sun. The structures with high density variation
appear restricted in both latitude and longitude and are located
relatively near the solar equator. The most significant difference
in the tomography result shown in the data projected to the solar
surface or to 1 AU is the longitudinal shift of the structures
observed. The most dominant of these structures map to regions
which appear bright in X-rays or Fe XIV observations, regions
of high solar activity. There is little or no evidence that the
regions of high density variation follow the heliospheric current
sheet (Fig. 7) other than in some general way. The regions of
low density variation are nearly all within the largest coronal
hole regions. Because of the comparison with IMP spacecraft observations,
and the relationship derived between the density variation and
density, we presume that we can map the general heliospheric density
values as in Fig. 7 as well as the density variations in three
dimensions over the whole of the heliosphere.
We will continue to upgrade our tomographic analysis techniques
as additional sources, more rotations and newer solar wind models
are incorporated. No attempt was made to select the correlated
data with respect to those features in the solar wind we know
to reoccur from one rotation to the next. We note also that the
mapped plasma parameters at 1 AU trail those from IMP by about
0.75 days, and if this were not the case the correlations between
the two data sets (especially the densities) would be markedly
improved. The restriction that corotation of the various structures
is maintained, is only one of computational convenience and numbers
of observational views. Certainly, different perspectives views
of any given structure are enhanced by corotation, but different
perspective views are also present simply from the outward motion
of the material within each structure. Thus, with enough observations,
the tomography could be performed in a time-dependent way so that
transient variations could also be mapped in the modeled data.
Acknowledgments.
We thank G. Woan of the Mullard Radio Astronomy Laboratory
(now at the University of Glasgow, Scotland) for making available
the Cambridge IPS data, and for his support of our analyses of
these data. The IMP data was made available to us with the help
of A.J. Lazarus and the MIT Space Plasma Physics Group. The work
of B.V. Jackson and P.L. Hick, was supported at the University
of California at San Diego by grant AFOSR-94-0070. The work of M. Kojima
was supported by the Scientific Research Fund of the Ministry of Education,
Culture and Science (grant 07454115). B. Jackson would
like to express his sincere appreciation to his colleagues and
the staff of the STElab, in Toyokawa City, Japan for supporting
this work during his recent stay as a visiting guest professor.
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