Originally published in: Solar Wind, NASA sp-308, 365-374, 1972
) from the
shock in the solar wind [Fairfield, 1969; Russell et al.,
1971] . Here we examine the extent that these problems
have affected measured interplanetary spectra, and in
this light construct a meaningful typical interplanetary
spectrum. Finally, we shall make some comments on the
interpretation of interplanetary power spectra, mainly as
a guide to the uninitiated.
of that at the surface of the earth, one of the first
questions that might be asked is whether the inherent
noise levels of the magnetometers are comparable to the
noise being measured. In particular, we would like to
compare the spectrum of the instrument noise with the
measured interplanetary spectrum. Unfortunately. very
few magnetometer experimenters have made this comparison.
Usually, at best, an rms amplitude noise over
some bandwidth is quoted with no clue as to the
distribution of the noise over this band.
Figure 1 shows
instrument noise levels for the OCO-3 search coil
magnetometer [Russell et al., 1970] and for two
fluxgate magnetometers, which were backup units for
two different deep space missions. Also on this figure is
a curve representing a typical quiet spectrum of one
component of the interplanetary field in this frequency
range. The justification for this curve will be given later.
 |
Figure 1. The instrument noise levels of three magnetometers.
Fluxgates A and B were not flown aboard
spacecraft but served as backup units for space missions
The search coil noise level is that of the OGO-3
instrument [Russell et al., 1970]. The dashed line is an
extrapolation of the quietest interplanetary power spectrum obtained by Siscoe
et al. [1968] assuming an f
spectral dependance.
|
We see that the noise spectrum of both fluxgate
magnetometers has a frequency dependence proportional
to l/f whereas the search coil noise has a
1/f
spectral dependence. The inverse cube dependence of
the search coil noise results from a 1/f noise in field
derivative units (nT/sec)
/Hz
(the search coil measures the
derivative of the field), which is converted to field units
nT/Hz
by dividing by
(2
f),
wherefis the frequency in
Hertz. The interplanetary power spectrum, however is
proportional to 1/f.
Due to their different slopes all
three spectra cross. The flux gates intersect the quiet
interplanetary spectrum at about 1 Hz. Thus, to the
extent that these are representative of fluxgate magnetometers in
actual operation in space, fluxgates should
be able to measure the quiet interplanetary spectrum to
about 1 Hz. Since The search coil noise spectrum crosses
the fluxgate noise spectrum at about 4 Hz, the search
coil is the better high-frequency instrument. We note.
however, that the search coil does not intersect our
hypothesized quiet interplanetary Spectrum until 15 Hz.
Thus, to the extent that fluxgates A and B are typical of
fluxgates used on actual missions we would expect that
the quiet interplanetary spectrum from 1 to 15 Hz
remains unmeasured. We shall see in fact our assumption
for the quiet interplanctary spcctruln must not extend
to frequencies much above 1 Hz and in reality the entire
interplanetary magnetic spectrum above 1 Hz remains
unobserved. In addition there are other sources of noise
which limit the detection of the interplanetary spectrum
at lower frequencies.
where K is integral. Then, it is easy to show that the
measured power spectrum P'(f) after aliasing is:
P'(f) = P(f) +
where 0
In the vicinity of the Nyquist frequencies of the
instruments listed in Table 1
, the typical spectrum of the
interplanetary field has at least a 1/f
Figure 3
shows two power spectra of the same time
series with digital windows of
10
the spectrum of the coarsely digitized signal deviates
strongly from the finely digitized spectrum at high
frequencies and approaches the line expected if the noise
were spread evenly across the band
(D
Figure 4 shows the difference between these two
power spectra on a semi log plot together with a second
test case. The second test case was performed on a
different random time series but with the same spectral
shape and integrated power. However, for the second
case, the coarse digital window was set at 0.25
nT. This
coarsely digitized time series had 705 digital steps in the
2048 points. The circled points indicate that the coarse
digitization spectrum was less than the fine digitization
spectrum. We see that at high frequencies the formula
D
In Table 1 we ranked magnetometers by the amount of
spectral folding present. Table 3 also ranks these
instruments but this time by their digital noise level as
calculated from (D
line for each of the magnetometers listed in
Table 3,
from the magnetometer's Nyquist frequency to our
assumed typical quiet interplanetary spectrum. If the
interplanetary field were "quiet" and digital noise were
the only noise source present in the measured spectra,
then we would expect that spectra derived by these
various magnetometers would follow the dashed line at
low frequencies and asymptotically approach the hori-
zontal noise line given for that magnetometer at high
frequencies. Naively, we could interpret the length of
the horizontal lines as the amount of wasted telemetry
during quiet conditions in the interplanetary medium.
However, conditions are not always quiet nor do all
these spacecraft remain solely in the interplanetary
medium. Figure 5 does not show
the noise level of the
UCLA OGO-5 fluxgate magnetometer at its highest
telemetry rate because from Figure 1
we expect that the
instrument noise level is greater than the quiet interplanetary
spectrum above 1 Hz. Similarly, the digitization
noise level of the OGO-1 and 3 search coil magnetometer
is not shown because its instrument noise level is greater
than its digital noise level except in its two low gain
states.
Figure 6 shows a power spectrum obtained in the
interplanetary medium by the OGO-1 search coil magnetometer
[Holzer et al., 1966]. We see that it is at least
a factor of 2 higher than the noise level at the similar
instrument on OGO-3 but has a very similar slope. The
increased power at low frequencies could be due to
interference near the OGO-1 spin frequency of 0.08 Hz.
which was not present in OGO-3 search coil magnetometer data.
Besides the possibility that this spectrum
simply shows the noise level of the magnetometer, there
is the possibility that this spectrum was contaminated by
bow shock associated waves. This spectrum was obtained
near the earth (the apogee of OGO-1 was 24 R
Figure 7 shows power spectra of the three components
of the interplanetary magnetic obtained by the UCLA
OGO-5 fluxgate magnetometer in solar ecliptic coordinates
calculated from 24,500 points with 500 degrees of
freedom. At the time of this measurement, the interplanetary
field measured upstream from the bow shock by
both Explorer 33 and 35 magnetometers had a roughly
constant solar ecliptic longitude of 260o and a latitude
that varied from about 0o to 20o. Using this orientation
and extrapolating from the OGO-5 solar ecliptic position
of (9.5, -10.8, 15.2) the field line did not intersect the
average position of the bow shock.
Thus, it is reasonable to assume that this spectrum is
unaffected by the presence of upstream waves and
represents a true interplanetary spectrum at 1 AU.
However, one spectrum cannot be considered "typical"
and further work is being undertaken to establish what
the typical spectrum is. However, we note that during
this spectrum the solar wind velocity was approximately
400 km/sec and the density was 3.3 cm
Table 1. The Nyquist frequency (half the sampling
frequency), the upper frequency cutoff of the instrument's
passband, and the ratio of these two frequencies
for spacecraft probing the interplanetary medium.
Nyquist Upper cutoff
Spacecraft frequency, frequency Hz ratio
IMP 1, 2, 3 0.025 5.0 200
IMP 4, 5 0.20 12.0 60
Explorer 33, 35a 0.10 5.0 50
Mariner 2 0.0135 0.33 24
Pioneer 6, 7, 8 0.33 5.0 15
Mariner 5 0.119b 0.80 6.7
Mariner 4 0.159b 0.80 5.0
Pioneer 9 1.7 1.7 1.0
Explorer 33, 35c 0.08 0.05 0.63
OGO 5d 0.43 0.22 0.50
3.54 1.77 0.50
27.8 13.9 0.50
OGO 1.3e 2.2 0.8 0.36
. 17 0.8 0.05
139 70 0.50
a GSFC magnetometer
b Calculated from average sample rate for high telemetry rate
c Ames Research Center magnetometer
d UCLA Fluxgate magnetometer
e Search coil magnetometer
[P(2nf
- f) +
P(2nf + f)]
f
f, f
is the Nyquist frequency and P(f)
is the true spectrum at frequency f.
Figure 2 shows the
result of aliasing for a magnetometer with R = 51 for 1/f
and 1/f
spectra. We see that the effect of aliasing is
significant over the whole frequency band for the 1/f
spectrum but is significant from only about 0.5
f to f
for the 1/f
spectrum. Table 2 lists the additional power
added to the spectrum at various frequencies for these
two spectra and a 1/f
spectrum. We note the aliasing
always at least doubles the power at the Nyquist
frequency .
Figure 2. The effect of aliasing on digitally sampled
time series for a spectrum proportional to f' and one
proportional to f
when the Nyquist criterion f or
reconstructing the spectrum is violated. In this example
signals up to 10 Hz were allowed to fold into the
analysis band of 0-0.2 Hz.
Table 2. The additional power added to the true spectrum by aliasing,
for a magnetometer whose bandwidth
is 50 times greater than the analysis band allowed by the
sampling rate, for three spectra with slopes proportional
to f
, f and f
Additional Folded Power
f/f
1/f, 1/f, 1/f,
percent percent percent
0.01 3.8 0 0
0.10 38 0.9 0
0.25 96 5.0 0.5
0.50 195 23 5.0
0.70 279 33 19
0.80 323 73 35
0.90 370 103 62
1.00 420 145 110
dependence. Thus
aliasing should only be a problem during typical interplanetary
conditions in the vicinity of the Nyquist
frequency. However, the instrument noise levels for the
flux gate magnetometers shown in Figure 1
had 1/ f
dependences. This noise is subject to aliasing, too.
Examining Table 2 we see that the noise level at the
Nyquist frequency was raised by a factor of 5 in this
example due to aliasing. Thus, for those instruments
with much larger upper cutoffs than Nyquist frequencies
we must reinterpret our conclusions about their capability
to resolve the quiet interplanetary spectrum.
Examining Figure 1,
we see that this spectral folding of
the instrument noise would limit them to measurements
below from about 0.1 to 0.2 Hz.
DIGITAL NOISE
The process of digitizing the magnetic field data adds
further noise to the spectrum. This noise is subject to
aliasing also, but in this case the folded power cannot be
removed in the instrument design. The ring noise due to
the digitization process can easily be shown to be
D/ 12,
where D is the size of a digital window [Bendat and
Piersol, 1966]. To understand how this noise affects a
measured spectrum, however, we must determine how it
is distributed over the power spectrum. The most
straightforward way to accomplish this is to compare
two spectra of the same time series: one digitized with a
large digital window, and one digitized with an extremely
small digital window. For this purpose, time
series were generated with a random number generator
with a Gaussian distribution of amplitudes. These time
series were then filtered with a digital single section
low-pass filter with a corner frequency below the
analysis band. The time series thus formed had a
1/ f
power spectrum.
nT
and 0.5nT. The
spectra were computed from 2048 points with 50
degrees of freedom. We note that there were only 315
steps between digital windows in the coarsely digitized
time series. The horizontal line is the power spectral
density to be expected if the digitization noise of
0.021 nT
were spread uniformly over the analysis band
of 0.5 Hz. The most obvious feature of
Figure 3
is that
Figure 3. Two spectra of the same time series: one
sampled with a digital window of 0.5 nT and one sampled
with a digital window of 10
nT. The original spectrum
was proportional to f.
The horizontal line is the power
expected if the rms digital noise
D/ 12 due to the
digital window (D = 0 nT) were spread uniformly over
the band 0 to f
where f is the upper frequency of the
analysis band the Nyquist frequency
/
12 f). A
surprising feature is that at low frequencies the coarsely
digitized power spectrum is less than that of the finely
digitized power spectrum. In other words, at low
frequencies the digitization process has consistently
reduced the power, whereas at high frequencies it has
raised the power. This is not a chance event, but has
been observed in every test case.
/
12 f is a good predictor of the digitization noise
added to the spectrum, but that at low frequencies
digitization actually reduces the power spectral density.
Figure 4. The difference between two pairs of-spectra
of a random time series with an f
spectral shape. One
spectrum of each pair had a digital window of 10 nT.
The other spectrum had a digital window of 0.5 nT in the
first case and 0.25 nT in the second case
/
12f).
This table is not as revealing
as it may seem at first, because a digital noise level for a
magnetometer with a high Nyquist frequency, will alter
the apparent shape of the spectrum much more than the
same noise level for a magnetometer with a low Nyquist
frequency. This is illustrated more clearly in
Figure 5
where the digital noise level is plotted as a horizontal
Table 3. The digital window Nyquist frequency and
digital noise level for spacecraft probing the interplaneary
medium the digital noise level given assumes that
the digital noise is spread uniformly across the analysis
band
Digital Nyquist Digital
window, frequency; noise level,
Spacecraft nT Hz nT
/Hz
___________________________________________________________
Mariner 2 0.7 0.0135 3.02
IMP 1,2,3 0.8 0.025 2.13
Mariner 4 0.7 0.159a 0.26
Explorer 33b 0.5 0.10 0.21
Explorer 0.4 0.08 0.17
33,35C
Mariner 5 0.4 0.119 0.11
IMP 5 0.4 0.2 6.7 
10
Pioneer 6 0.5 0.33 6.3 10
IMP 4 0.32 0.20 1.3 10
Explorer 35b 0.19 0.10 3.0 10
Pioneer 7,8 0.25 0.33 1.6 10
Pioneer 9 0.40 1.75 7.6 10
OGO 5d 0.125 0.43 3.0 10
0.125 3.47 3.8 10
0.125 27.78 4.7 10
a Calculated using average sample rate at highest telemetry rate
b GSFC magnetometer
C Ames Research Center magnetometer
d UCLA flux gate magnetometer
Figure 5. The digital noise level and Nyquist Frequency
of a majority of the magnetometers which have
measured the interplanetary magnetic field. The horizontal
line marks the digital noise level. The dot marks the
Nyquist frequency. The dashed line shows the expected
quiet interplanetary spectrum.
MEASUREMENT OF THE INTERPLANETARY
FIELD NEAR THE EARTH
It is now well established that near the earth but
upstream from the bow shock, waves are present in the
interplanetary medium that are not present far from the
earth [Fairfield 1969; Russell et al., 1971] While these
waves do not apparently affect the average magnetic
field strength [Fairfield 1969], they do increase the
power in the frequency range from 10
to 1 Hz.
Although the waves are seldom present unless the field
line simultaneously threads both the satellite and the
bow shock, it is difficult in practice to determine
whether a particular field line intersects the shock
because of the variability of the position of the shock
and the uncertainty in the direction of the field due to
the presence of the waves. Thus, even when the
orientation of tile interplanetary field is known, care
must be exercised in the interprelation of interplanetary
spectra obtained near the earth. However, from
Figure 5.
we see that to extend our knowledge of the interplanetary
spectrum much above 0.1 Hz we must examine
near-earth data.
), and no
data were available on the orientation of the interplanetary
magnetic field at this time. Furthermore, this
spectrum differs significantly from others measured in
the same region.
Figure 6. A comparison of the OGO-1 search coil
measurement of the spectrum of the interplanetary
magnetic field [Holzer et al, 1966] with the noise level
of a simlilar instrument on board OCO-3 [Russell et al.,
l970].
Figure 7. Power spectra of the three solar ecliptic
coordinates of the interplanetary magnetic field
obtained by the UCLA OGO-5 fluxgute magnetometer
from 2123 to 2230 UT on March 7, 1968. At this time,
the field line through OGO-5 did not intersect the
expected position of the bow shock. The expected
instrument noise level, the digital noise Ievel, and their
sum are also shown.
and both
quantities were changing only slowly over the course of
the day [J. Binsack, private communication 1970]. In
other words, the solar wind was average during this
period of time.
POWER SPECTRAL DENSITY OF THE
INTERPLANETARY MAGNETIC FIELD
Having discussed the possible errors in the measurement
of the interplanetary power spectrum, we will now put
together what we feel is the best estimate of the
interplanetary power spectrum. This is shown in Figure
8. At the lowest frequencies
(10
to 10
Hz) to
define accurately the power spectrum requires continuous
data in the interplanetary medium for many days.
Earth orbiting spacecraft cannot acquire such continuous
data. Of the two series of interplanetary probes.
the Mariner series and the Pioneer series spectra have
been published for the lowest frequencies only for the
Mariner 2 [Coleman, 1968] and Pioneer 6 data [Sari and
Ness, 1969]. However the normalization of the power
spectra of Sari and Ness [ 1969] are obviously incorrect
and so we have used the Mariner 2 data in
Figure 8.
Mariner 2 was launched during a very active period of
time and inward toward Venus. These two effects would
tend to increase the power observed and indeed the
Mariner 2 curve appears to be somewhat high. We note
that since Tables 1 and
2 indicate a possible spectral
folding problem, we have plotted the Mariner 2 data
only to one-quarter of its Nyquist frequency.
 | Figure 8. A composite spectrum of the radial component of the interplanetary magnetic field as observed on Mariner 2 [Coleman, 1968], on Mariner 4 [Siscoe et al., 1968], and on OGO-5. Three spectra showing the range of variability of the interplanetary spectrum are shown for Mariner 4. Since the Mariner 2 data are consistently higher than the Mariner 4 data in the overlapping range of frequencies, it is assumed that the Mariner 2 data were obtained during an unusually disturbed period of time, and the typical spectrum has lower power. Three straight line segments have been drawn with slopes of -1, -1.5, -2 to roughly represent the expected average spectrum near 1 A U. |
At the intermediate frequencies thc Mariner 4 [Siscoe
et al., 1968] data have been used because these data
have been analyzed to show the range of variability of
the spectrum. Active, intermediately active and quiet
spectra are shown. We note that the Mariner 4 spectra
asymptotically approach a value of about 0.1
nT/Hz at
high frequencies, which is a factor of 2 lower than our
estimate of the digitization noise in Table 3. This is
possibly because the Mariner 4 data samples are not
equispaced as we have assumed in the calculation of the
digital noise level. Finally, at the highest frequencies we
have used the OGO-5 power spectrum shown in Figure 7,
which as suggested in the previous section, appears to be
typical of average solar wind conditions since it joins
smoothly with the intermediate activity spectrum of
Mariner 4. We note that although we have chosen to plot
only the power in the radial component on this figure,
the other components have similar spectral form.
On this figure we have drawn three straight lines
with slopes of -1, -1.5, and -2 with changes in slope
occurring at 3 10
and 10 Hz. We see that these
straight lines are roughly parallel to the spectrum in the
three frequency ranges. The two breaks in tile spectrum
are somewhat arbitrary, however, and Sari and Ness
[1969] claim that the break between f
and f
occurs at about 5 10
Hz. However, this is not clear
from their data since they present no spectra that cover
the region of their hypothesized change in slope.
Since Russell et al. [1970] showed that power spectra
obtained in the interplanetary medium from 1 to 140 Hz
with the search coil magnetometer were at the instrument's
noise level, there must be a further increase in the
slope of the spectrum possibly from
f to J
above
1 Hz. However, no other limits on the possible spectrum
above 1 Hz can be determined with the present data.
and a magnetic field of 5 nT is
streaming past a spacecraft with a velocity of 6 times the
Alfven velocity. Thus, waves propagating in the solar
wind are severely Doppler shifted. The amount of
Doppler shifting depends on the size of the component
of the solar wind parallel to the phase velocity of the
wave. If a wave is propagating perpendicular to the solar
wind velocity, therefore, the Doppler shifting is zero.
However, if a wave with phase velocity less than that of
the solar wind (most electromagnetic waves under
typical solar wind conditions) is propagating parallel or
antiparallel to the solar wind it will be severely Doppler
shifted. Waves propagating parallel to the solar wind will
be Doppler shifted to higher frequencies maintaining
their sense of polarization, and waves propagating
anti parallel to the solar wind will be Doppler shifted to
lower frequencies if their phase velocity is greater than
half the solar wind velocity and to higher frequencies if
their phase velocity is less than half the solar wind
velocity. In both antiparallel propagation cases, however,
the wave polarization observed in the satellite frame is
reversed unless the phase velocity is greater than the
solar wind velocity. In short, then, the fact that the solar
wind is flowing past the observer and is, in fact, usually
super Aifvenic and supersonic, mixes the power spectrum
of the signal in the plasma rest frame as well as
mixing cross correlations between components. Thus, it
is not simple to interpret these power spectra.
To understand the physical processes occurring in the
magnetic field, such as which wave modes are present, it
is essential to perform cross correlations with other
plasma parameters.
At present there is some controversy as to the
importance of discontinuities versus waves in determining
the interplanetary power spectrum [Sari and
Ness 1969; Belcher et al., 1970] . Step functions in the
magnetic field, whether they are propagating as waves or
whether they are simply convected with the solar wind
velocity, will both contribute to a
1/f spectrum at high
frequencies. (We note that the low-frequency spectrum,
below approximately the frequency corresponding to
the average spacing of the discontinuities, need not be
proportional to 1/f ) Furthermore, there is no necessity
that the natural wave spectrum between discontinuities
not be proportional to 1/f.
Thus, the spectral shape of
the interplanetary spectrum provides no simple answer
to this controversy. To distinguish between propagating
and non propagating structures requires examination of
both the field and plasma behavior. We note the
anisotropies in the solar wind plasma distributions
further complicate these identifications [Hudson,
1970] .
From our test cases, it appears that digital noise is distributed uniformly over the power spectrum at least at high frequencies. However, at low frequencies digitization actually reduced the power. Although this undoubtedly altered the power spectra obtained in the interplanetary medium, its effect is small (Fig. 3) and cannot account for the observed changes in slope. Aliasing could be a problem in the creation of power spectra from the data for many of the interplanetary magnetometers. However, due to the observed natural spectrum of interplanetary fluctuations this should only be a serious problem for frequencies above one-half the Nyquist frequency.
Although the interplanetary spectrum near the earth
can be contaminated by waves associated with the
earth's bow shock, we can combine OGO-5 data with the
Manner 2 and 4 interplanetary spectra to create the
spectrum from about 5
10 Hz to 1 Hz, if care is
taken to exclude times when the magnetic field line
threads both OGO-5 and the shock front. The spectrum
of the radial component is approximately proportional
to f
up to 3
10 Hz; then it is proportional to
f up to about
10 Hz; and finally it is proportional
about f
up to at least 1 Hz. It is quite probable that
the spectrum undergoes another change in slope above
1 Hz.
Finally, we stress the difficulty in interpreting the power spectrum of the interplanetary magnetic field by itself. Doppler shifting mixes frequencies and different physical processes can result in the same spectrum. Cross correlations with simultaneous plasma data are necessary. Multispacecraft studies could also be very fruitful.
Belcher, J. W.; Coleman, P. J., Jr.; Davis, L., Jr.; Jones, D. E.; and Smith, E. J.: Waves and Discontinuities in the Solar Wind. Space Phys. Preprint, California Institute of Technology,1970.
Bendat, J. S.; and Piersol, A. G.: measurement and Analysis of Random Data, Chap. 7, John Wiley and Sons, New York,1966.
Childers, D. D.; Russell, C. T.; and Coleman, P. J., Jr.: OGO-5 Observations of Upstream Waves in the Interplanetary Medium: Statistical properties, in preparation, 1971.
Coleman, P. J., Jr.: Characteristics of the Region of Interaction Between the Interplanetary Plasma and the Geomagnetic Field: Pioneer 5. J Geophys Res Vol.69,1964, p.3051.
Coleman, P. J., Jr.: Variations in the Interplanetary Magnetic Field: Mariner 2, 1, Observed properties. J. Geophys Res Vol 71,1966, p.5509.
Coleman P. J., Jr.: Turbulence, Viscosity and Dissipation in the Solar Wind Plasma. Astrophys J Vol.153,1968,p.371.
Fairfield, D. H.: Bow Shock Associated Waves Observed in the Far Upstream Interplanetary Medium. J Geophys Res Vol 74,1969, p.3541.
Holzer, R. E.; McLeod, M. G.; and Smith, E. J.: Preliminary Results from the OGO-1 Search Coil Magnetometer: Boundary positions and magnetic noise spectra. J. Geophys Res Vol 71, 1966, p. 1481.
Hudson, P. O.: Discontinuities in an Anisotropic Plasma and their Identification in the Solar Wind. Planet Space Sci Vol 18,1970, p.1611.
Ness, N. F.: Magnetometers for Space Research, Goddard Space Flight Center, Report X-690-70-78, 1970.
Russell, C. T.; Holzer, R. E.; and Smith, E. J.: OGO-3 Observations of ELF Noise in the Magnetosphere, 1. The nature of the equatorial noise. J. Geophys Res Vol.75,1970, p.755.
Russell, C. T.; Childers, D. D.; and Coleman, P. J., Jr.: OGO-5 Observations of Upstream Waves in the Interplanetary Medium: Discrete Wave Packets. J. Geophys. Res., Vol.76,1971, p. 845.
Sari, J. W.; and Ness, N. F.: Power Spectra of the Interplanet;lry Magnetic Field. Solar Phys Vol 8, 1969, p.155.
Siscoe, G. L.; Davis, L.; Coleman, P. J., Jr.; Smith, E. J.; and Jones, D. E.: Power Spectra and Discontinuities of the Interplanetary Magnetic Field: Mariner 4. J. Geophys. Res., Vol. 73, 73,1968, p.61
nT. The original spectrum
was proportional to f.
The horizontal line is the power
expected if the rms digital noise
D/ 12 due to the
digital window (D = 0 nT) were spread uniformly over
the band 0 to f
where f is the upper frequency of the
analysis band the Nyquist freauency
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