Location
SRBL locates bursts on the sun's disk using a novel technique that exploits spectral artifacts
introduced by the geometrical response of its antenna and feed.
Scheme
The sun's radius is 16 arcminutes, as seen from Earth, and SRBL tracks it using a parabolic dish
with a focal length of 27 inches. If this reflector had an optical-quality (mirrored) surface, then a solar
image 6.4 mm in diameter would appear at its focal plane. But the actual surface is relatively rough and
astigmatic, causing such blurring that the focus is instead only heated by incident radiation. At radio
wavelengths (of 1 cm or more) SRBL's antenna is smooth and well-formed, but solar images are still
blurred, this time by diffraction. The limiting angular resolution is W  /2D radians, where is
wavelength and D is dish diameter; i.e. a point source on the sky focuses to a nearly Gaussian smear with
an intensity that varies with angle as I()  exp(−2/2W2). With D = 6 ft, then W  5/, where  is
frequency in GHz; i.e. SRBL's "beam pattern" nearly matches our sun’s apparent radius at the highest
operational frequency of 18 GHz. Thus, so long as SRBL points near sun-center, it can maintain a view of
the entire solar disk at all frequencies while taking data.
Figure 1. Overall geometry.
Suppose a dish 50 feet in diameter was used instead. A beamwidth of 18' at 2 GHz would obtain,
enough to include the whole sun. Solar monitoring might proceed at 2 GHz and, when a burst is noticed,
mechanical scans could begin while more frequencies are sampled. This may take several tens of seconds,
during which time the eruption would evolve, or end. Location resolutions of 2' might be reached,
corresponding to beamwidths around 18 GHz. Then again, such higher-frequency components would often
be missed. SRBL avoids this shortfall by using a smaller dish and sequencing through a large set of
frequencies every few seconds, each one poised to observe activity occurring anywhere on the sun.
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Figure 2. Left: solar "images" at focus for different wavelengths,
shown actual scale. Right: cross section of SRBL beam pattern.
SRBL's circular receiving element, or “feed” is 7 inches across, and therefore blocks only ~1% of
the incident radiation. It is coaxial with the dish's axis of symmetry, or “boresite” and located with its front
face on the focal plane. That face is clad with two interleaved copper spirals spanning radii between 0.15
cm and 7.5 cm, and that vary in radius with angle roughly as r ~ (7.5cm) e −n/6, where n counts the number
of turns. It is receptive to frequencies from about 500 MHz to just over 18 GHz.
36
Figure 3. Interlaced traces on the front face of SRBL's log-spiral feed, drawn to scale.
Equal areas are covered and uncovered. Traces at the perimeter (7.5 cm from center,
not shown) taper in width. This pattern is backed by a cylindrical cavity, 6 cm deep,
with dielectric inside shaped to enhance responses for radiation coming from in front.
This feed is a quarter-wave, circularly polarized receiver; i.e. electromagnetic radiation impinging
upon those traces is absorbed preferentially at radii of r ~ /4. However, since logarithmic spirals do not
form a set of nested circles, this resonant response is not centered, but instead follows geometrically at radii
of R ~ −dr/dn ~ /24, roughly matching the sun's diameter at 2 GHz. In other words, the effective electrical
center of SRBL's receiver is offset from its boresite, spiralling around at increasing radii for larger
wavelengths. As a result, measured spectra contain artifacts, “modulations” that vary in phase and
amplitude depending upon where a radio source is located. For example, suppose an event occurs on the
sun's central meridian, but at 20 north latitude; i.e.  = 16'sin20 = 5.5' above the boresite (assuming
perfect pointing at sun-center). A corresponding burst "image" appears at a radius of R = (27 inches) tan
= 0.11 cm on the feed, where wavelengths near  ~ 24R = 2.62 cm ( ~ 11.5 GHz) encounter maximum
reception. Half a turn later around the spirals, at ' ~ 2.62 e0.5/6 = 2.85 cm, the effective electrical response-
37
center will be offset by   tan−1[( + ')/24F] = 11.4' from this image. Mean beamwidths at ' are W 
25.6', and so the observed flux is reduced to exp(−2/2W2)  90.5% of maximum there. Further dips
occur with each spiral cycle, regularly spaced in log.
There is no frequency at which signals reach maximum strength for on-axis sources. Modulations
vanish there, but fluxes are uniformly reduced by R2/2W2  2%. At all other burst locations these artifacts
should appear across the entire spectrum, exhibiting consistent phases and amplitudes within every period.
SRBL “locates” bursts by deconvolving such artifacts.
Figure 4. Left: position of effective electrical center of SRBL's feed, shown at actual scale.
Right: separate center offsets in azimuth and elevation versus the logarithm of frequency.
These are 90 out of phase and sinusoidal, decreasing in amplitude for higher frequencies.
Operational Brief
First, calibrations are needed to account for non-ideal circumstances: asymmetries, non-linearities,
etc. These provide an instrument-specific set of parameters with which to fit spectral modulations.
There is a difficulty however; only the observed spectra are available, which include modulations.
The shape of incident signals are unknown in principle, yet must be reconstructed. If those inputs varied
rapidly with wavelength, SRBL's task would be hopeless. But solar bursts at microwave frequencies are
typically broadband, smoothly extending over many GHz, peaking in intensity roughly between 2 and 10
GHz. As many as thirteen modulations would appear in spectra spanning 2 − 18 GHz. This is neither too
many nor too few to allow use of smoothness criteria in extracting event locations, if at least a handful of
frequencies are sampled within each period.
To obtain reasonable signal-to-noise ratios, SRBL dwells 40 ms to measure fluxes (actually
integrating only during the later 30 ms), requiring 4.8 seconds to sample 120 frequencies: at most 5 in each
of the 245, 410, and 610 MHz bands, and the rest logarithmically spaced from 1 to 18 GHz. Microwave
bursts often evolve on similar time scales. Therefore, complete spectra can usually be recorded quickly
enough to avoid time-dependent distortions of modulations.
The quiet-sun is continuously observed, and levels are averaged back in time to smooth over slow
gain drifts or weather changes. These signals provide reference spectra and locations. When certain
thresholds are crossed, bursts are declared, and the excess-spectra are separately analyzed. Burst locations
with respect to sun-center are given by the vectorial difference between excess and quiet-sun offsets.
Noise and other irregularities can severely limit capabilities. There is, however, considerable
flexibility in implemention strategies for sampling, calibration, and software algorithms. (A number of
different techniques have been tested, and ideas are still evolving.) Overall, SRBL's design is balanced and
well matched to its intended signals.
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Figure 5. Relative intensity versus frequency for different burst locations:
1) at sun-center, 2) half way to East limb, 3) at East limb, 4) at West limb.
Such modulations appear throughout burst and quiet-sun spectra, forming
the basis of SRBL's location capability.
Algorithms
SRBL operates in Cartesian coordinates referenced to its antenna boresite, with angular units. The
effective electrical center of its feed is (X,Y) = (Rcos,Rsin), where  is the spiral angle. (See Figure 6
for layout details.) A point source on the sky at location (x,y) = (rcos,rsin) with spectrum S0() will
produce measured fluxes of S() = S0()exp(−2/2W2) where 2 = (x−X)2+(y−Y)2. Modulations appear in
S due to inbedded  dependencies, which increase in amplitude with  and r.
Suppose (X,Y), W, and  are known as functions of frequency through calibrations to be
described later. The task is to extract (x,y), given S. S 0 is not known, but it is presumed to be “smooth” on
scales comparable to instrumental modulations. Therefore the function f = lnS + [(x−X) 2+(y−Y)2]/2W2
must be unmodulated.
39
Figure 6. Focus geometry.
There are several means to obtain (x,y), each prone to different sorts of error. The present option
of choice is to construct a grid of all likely (x,y) values, calculating f at each point, and selecting the best
fit. This fit is recognized in one of two ways depending upon which spectrum is being examined. For
quiet-sun spectra, which yield pointing offsets that are in place before bursts occur, f is approximated by a
quadratic in frequency (anticipating a black-body distribution of fluxes). For burst spectra, smoothness is
indicated when no residual modulations are found.
In more detail, consider burst data within one modulation period, given by a change in  of 2.
Label the frequencies in that span with indices, n. Compute fn = lnSn + [(i−Xn)2+(j−Yn)2]/2Wn2 at each n,
with a grid spacing of  = 1 arcminute and where (i,j) are held as fixed integers ranging from −32 to +32.
Now suppose these fn are to be fit by aun+bvn+c where (u,v) are known functions (X,Y)/W 2, and (a,b,c)
must be determined.
Proceed by minimizing the total variance, e = Σ[fn − (aun+bvn+c)]2wn/Σwn, where relative weights
of wn = exp(−(n−0)2/42) are used to emphasize frequencies in the middle of this period (at 0). Setting
partial derivatives with respect to a, b, and c to zero gives FU = aUU + bUV + cU, FY = aUV + bVV + cV,
and F = aU + bV + cw, where new symbols have been defined: G ≡ Σgnwn, GH ≡ Σgnhnwn, and w ≡ Σwn.
Therefore, a = (VfuVvv − VfvVuv)/d, b = (VfvVuu − VfuVuv)/d, and c = [F(UU.VV − UV.UV) − U(UF.VV −
UV.FV) − V(UU.VF − UV.UF)]/d, with d = VuuVvv − VuvVuv and Vgh ≡ GH − G.H/w. Then revisit all n to
eliminate measurements at which aun+bvn+c differs from fn by more than 5√e, and repeat the whole process
until a good fit is returned within this span of frequencies. Note, V gh is invariant with respect to commonmode offsets in g or h, so the results (a,b,c) are stable.
Best (i,j) grid points correspond to the least average (a 2+b2), accumulated for periods centered at
every frequency. These are refined by fitting Σ(a2+b2) with a quadratic in (x,y) extending to nearby points,
and minimizing.
40
To assess location uncertainties, approximate f − (au+bv+c) by A+B within each period, and use
the overall fit variance to weight [(x−X)2+(y−Y)2]/W4. Then sum these for all frequencies in the spectrum,
invert, and take the square root.
For quiet-sun data, choose (u,v) = (,2) and fit 8 spans of . Within each span, use weights of
wn = exp(−(n−0)2/322). Best (i,j)’s locate the least average variance accumulated over all spans.
Otherwise, everything proceeds as for burst data.
Example
Three spectra are needed to extract event locations with respect to sun-center (xB−xQ,yB−yQ): SC
(cold-sky), SQC (quiet-sun plus cold-sky), and SBQC (burst plus quiet-sun and cold-sky). SC is obtained from
pre-dawn calibrations, and the others are measured during patrol. From these, separate boresite offsets are
calculated: SQ = SQC − SC yields (xQ,yQ), and SB = SBQC − SQC yields (xB,yB).
As an example, consider the data shown in Figure 8, taken at the peak of a moderately sized burst.
Note that cold-sky flux levels SC (designated 'C' in the top panel) are significant and evidence considerable
irregularities. Vagaries in SC, apparent also in SQC and SBQC, are removed by subtraction (lower panels).
Modulations are obvious in the resulting SB, indicating large values for xB and/or yB. Fitted offsets were
(xQ,yQ) = (−6.6',−2.3') ± 1.3' and (xB,yB) = (−21.7',2.8') ± 1.1', giving (xB−xQ,yB−yQ) = (−15.1',5.1') ± 1.7'.
Optical images taken through an H filter by SOON telescopes showed a flare at (−10.9',8.3') with
respect to sun-center. This is 5.3' distant from SRBL coordinates. Assuming an optical uncertainty of 1'
(to account for possible H vs. radio source mismatches) and adding errors in quadrature, a relative
discrepancy of 5.3/(1.72+12)1/2 = “2.7-sigma” is indicated. [By comparison, the radio spectrum recorded
by SRBL just after this burst crossed location threshold yielded (xB,yB) = (−19.4',3.8') ± 1.1', giving
(xB−xQ,yB−yQ) = (−12.8',6.1') ± 1.7', which is 2.9' distant, for a 1.5-sigma discrepancy. A tendency for
early spectra to yield better location accuracies has been noticed in several other bursts.] These results are
depicted in Figure 7.
Figure 7. Locations calculated from the spectra of Figure 8,
placed on a scale image of our sun. Error circles are shown
with respect to the boresite; uncertainties for (xB−xQ,yB−yQ)
must be added in quadrature from (xQ,yQ) and (xB,yB).
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Figure 8. A burst recorded at OVRO on 3 June 2000 at 19:19:07 UT. Upper panel: raw spectra for the Cold
sky, Quiet sun plus Cold sky, and Burst plus Quiet sun plus Cold sky. Middle panel: (B+Q+C) − (Q+C) as
measured and after removing fitted modulations. Lower panel: (Q+C) − C.
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Before discussing resolutions, further details will be presented of the algorithms used during patrol
and calibration. These include ways of averaging quiet-sun spectra, deciding when bursts have occurred,
and measuring modulation parameters.
Patrol
During patrol, the sun is held in view while SRBL's receiver cycles through 120 frequencies,
chosen among those for which gain factors exist. Around 85 of these are in the 2 − 18 GHz range, and are
somewhat logarithmically spaced. Attenuation is dynamic, to keep fluxes on-scale. A reference noise
diode constantly toggles on and off, doubling the time needed to acquire each complete spectral record.
Noise-diode signals are averaged with a 10 minute folding constant, allowing gain drifts to be tracked (at
every frequency independently) on time scales comparable to typical burst durations.
Quiet-sun plus cold-sky signals at each frequency n, SQCn are accumulated as weighted averages of
SQCn−1 plus the latest measured fluxes Sn: SQCn = [e−1/tSQCn−1+k(1−e−1/t)Sn]/(e−1/t+k−ke−1/t), where t counts the
number of records folded together (building from 0 to 60 as a patrol starts, then staying fixed, equivalent to
a ~10 min time constant), and k = 0, 1/2, or 1 depending on whether a burst is in progress, the solar state is
ambiguous, or a quiet sun prevails. The intensity variance V SCn is accumulated similarly. Status
assessment logic is as follows:
- If 0.85 < Sn/SQCn < 1.15 for >90% of the frequencies sampled, then the sun is quiet.
- If Sn−SQCn−Atm > 50 sfu for >30% of the frequencies sampled, then a flare is active.
- If Sn−SQCn−Atm > 500 sfu for >30% of the frequencies sampled, then a burst is active.
- If none of the above, then there is ambiguity. Also, if t < 60, then k is forced to unity and no events can
be declared until a “quiet” sun has been established. “Atm” is a contribution to S n−SQCn due to transient
atmospheric absorption: estimated as A3, where A is the average of (Sn−SQCn)/3 above 10 GHz. By
“burst” is meant an event large enough to try locating. As more records accumulate for each burst, a
weighted average of (xB,yB) is kept, discarding outlying data.
Calibration
Feed phases must be determined and (X,Y) & W measured beforehand. This neccesitates a
special run, which views the quiet sun from angles of 0, ±0.25, ±0.5, ±1, and ±2 degrees in azimuth and
elevation separately. The cold-sky is sampled at beginning and end with an elevation offset of +20 degrees,
chosen to be larger than 2W at 610 MHz. While this “trajectory” proceeds, SRBL's receiver cycles through
all 596 standard frequencies plus 4 YIG-switch pauses. (It takes 30 minutes to take data.) Attenuators are
fixed to particular values for different frequency ranges. The noise diode is held off. A usable dwell time
of 48 seconds per offset is kept, so 1200 measurements are taken, giving 2 samples to compare at each
frequency. These are corrected for pedestals, non-linearity, and attenuation based on parameters derived
from a just-prior gain calibration (which itself takes 30 minutes). Variances are computed. Outliers are not
eliminated.
During analysis, an interpolated cold-sky background is subtracted at each offset. Intensities
should therefore vary as exp[−(x−X)2/2Wx2]exp[−(y−Y)2/2Wy2], where (x,y) = (xQ+x',yQ+y') for
commanded offsets (x',y'), with y'=0 (x'=0) for azimuth (elevation) scans. (Note, limb brightening can be
ignored, since the beamwidth is so large.) The problem is then linearized, and signals S n corresponding to
angles An are fit by lnSn = a+bAn+cAn2, ignoring An where less than 1/4th of the maximum signal is
observed. Peak offsets (x,y) = (xQ−Xn,yQ−Yn) and beamwidths (Wx,Wy) are thereby obtained for every
frequency.
43
To separate (xQ,yQ) from (X,Y), frequencies are first converted to phases:  = 2[a+bln+c(ln)2],
with (a,b,c) derived from earlier measurements. Then successive groups of (x,y) spanning periods of  are
fit by (xQ+Cxcos+Sxsin, yQ+Cycos+Sysin). Mean W and (X,Y) are written to a plottable reference
file, returning plots like that in Figure 4, left-hand side. Frequencies with malformed Gaussians, too many
bad measurements, or outlying (X,Y) are eliminated. Estimated 2-dimensional signal peaks (with units of
channels) are also written to file, and later compared with concurrent quiet-sun fluxes reported by RSTN
(where units are in sfu), thus establishing gain factors.
Resolution
Recall that 2W2ln(S0/S) = (x−X)2+(y−Y)2. Treating S0, (X,Y), and W as well known parameters,
an error expression for x may be derived: (x−X) 22x ≈ W4(F/S)2, where F/S is the relative flux resolution,
which is typically around 5% for most frequencies between 2 and 18 GHz. Overall, x is found by
computing [∫218(S/x2)d/∫218Sd-1/2, includung a sum of inverse variances for all , weighted by S. As
an estimate, assume a flat spectrum and calculate ∫218dln/x2ln9. For comparable x and y, location
uncertainties are r ~ 5'(1+50r2)−1/2, with the offset r measured in degrees. In arriving at this expression,
approximations X  (1/)cos and W  5/ are used. On-axis sources (r = 0) give r ~ 5' per record,
whereas solar-limb sources yield about half that.
With pointing that is close to sun-center for at least 10 minutes (~60 records), Q ~ 0.6' errors may
be expected for pre-burst sun locations. However, 1' is always included as an estimate of systematic
uncertainty. Actually, since signal-to-noise ratios are usually worse for quiet-sun spectra, Q is often larger
than 1.5'. Adding this in quadrature with similarly figured burst location errors, B, yields final
uncertainties of 5' (or ~3') per record for sun-center (or solar-limb) events. More then one minute (>6
records) of accumulated good data should be needed to obtain 2' resolutions.
Location uncertainties depend directly upon flux resolutions and spectral shapes. More subtle
dependencies appear in the handling of calibrations, sampling, irregularities, etc. The following influences
degrade performance further.
SRBL's antenna is commanded every ½ second with new pointing coordinates, in step sizes of
1/500th of a degree. In calm weather, and with a rigid structure, such angular variations would translate to
tiny relative flux errors of 1−exp(−0.002 2/2W2) ~ 0.001% at ~10 GHz. This antenna jiggles a little with
every pulse of the motors, yet even ~1 mm motions at the feed (~5') introduce flux uncertainties of only
~1.4%. Wind can shake the structure much more violently. Such dishes act as effective sails, warping and
fluttering in heavy breezes. The Yagi catches gusts, jerking on the spiral feed. Errors rapidly worsen,
reaching 5% at 10 GHz for ~2 mm wobbles, effectively guaranteeing location uncertainties over 5'.
Slow effects like the passage of clouds and gentle rain do little damage. Sleet or lightening are
devestating however.
Electromagnetic interference takes many forms, always damaging. These include single spikes,
hash across several frequencies, broadband jumps, etc. A naive estimate would suggest that if 20% of the
spectrum were affected by EMI, then location uncertainties would only rise by a factor of 1.2 1/2. But since
x varies in proportion with W2, then modulation fits improve with the square of frequency; i.e. the upper
ends of burst spectra are more important.
As if to counter this trend, flux resolutions are found to be coarser at higher frequencies. In the
above example from 3 June 2000, spectra became more erratic above 10 GHz. This was due in part to the
poor characteristics of OVRO's prototype receiver; with gain factors exceeding 10 sfu per channel; i.e.
signal variations of a few channels resulted in wildy changing “measured” fluxes. In addition, the actual
burst spectrum apparently included a dip between 7 and 8 GHz, which is similar in width to modulations
there, confusing the fit a little.
44
Improvements
The following are possible ways to enhance SRBL location capabilities. These are categorized as
dealing with calibration or patrol data, and grouped by approach. Counter arguments are presented in
brackets. This is an ongoing research and development process; it is unlikely that every approach has been
contemplated.
1.
Calibrations:
1.1
Altering the trajectory or sampling scheme to improve data quantity and/or quality:
1.1.1
Sample more offsets. Alter the trajectory to reduce the distances moved, and dwell long enough to
acquire just one sample at each frequency. Keep the total elapsed time below 30 minutes to avoid weather
changes, gain drifts, etc. [But it is unclear whether, say, twice as many measurements, each half as precise,
will result in improved Gaussian fits.]
1.1.2
Take more samples at each frequency, to eliminate stray points. An additional 8 minutes would
provide a third sample at each offset, and allow some poor measurements to be recognized. [But we might
do as well by ignoring outlying values among the pair of measurements taken now.]
1.1.3
Perform gain stabilizations while taking data. [This increases acquisition time, and might not help
in any case.]
1.2
Avoid, fix, or eliminate irregularities in (X,Y) and W. Beyond measurement noise, (X,Y) and W
should vary slowly with frequency. Except for phase and minor amplitude adjustments, X and Y should
replicate one another and not differ when feeds are changed or later calibrations run. But irregularities are
observed:
outliers: single-point deviations from smoothness
wiggles: small, rapid deviations from (X,Y) sinusoids or W flatness
jiggles: period-doubled oscillations in X2+Y2, due to astigmatism in feed geometry/response
tilts:
uncomfortably significant amplitude differences between X and Y
Outliers are often due to bad data (e.g. poor measurements at particular offsets) which yield differences in
Gaussian fit parameters, but not bad enough to fall outside our tolerances. Wiggles may result from similar
causes. Jiggles are of unknown origin. Tilts could indicate that our feed is astigmatic, or not perpendicular
to the boresite axis. SRBL already smooths (X,Y) by fitting to Cxcos+Sxsin, but consider also:
1.2.1
Forcing (X,Y) and/or W to follow prescribed functional forms. [This was tried, and made matters
worse.]
1.2.2
Shifting phase and amplitude to compare X and Y. [The astigmatism may be real, however, and
should therefore be measured, not “corrected”.]
1.2.3
Comparing against “factory” templates which have all irregularities removed. Templates may be
obtained by combining calibrations taken with each particular feed. Subsequent calibrations might take
less time to determine only initial phases and systematic pointing offsets. There are likely to be no aging
effects of concern.
1.3
Other issues.
1.3.1
What damage is done by assuming a particular phase/frequency relationship? For any particular
feed, a calibrated correspondence should be OK. [Does the same parameter set characterizes other feeds?]
1.3.2
Smooth out 2D peak flux differences to get better gain factors. Peak fluxes display large
variations between frequencies. These may just be due to sensitivities in the Gaussian fitting algorithm.
[First, check if smoothing yields better solar flux precision.]
1.3.3
Do a cross correlation of peak spectra with the cold-sky and noise diode spectra taken during
calibration to recognize artifacts, outliers, etc.
45
1.3.4
Do a Fourier analysis of calibration spectra, and strip off components not at frequencies
corresponding to modulations.
2.
Patrol:
2.1
Guide or track differently
2.1.1
Lock onto sun-center using an external optical guider. If (xQ,yQ) = (0,0) was guaranteed, then
calculating burst locations would be simpler and more precise. Consider dynamic feedback in real time
using an external guider whose relationship to the boresite has been established. A lens and quadrant cell
might be used. [Clouds present an obvious difficulty. Signal assessment must be very sophisticated just to
deal with times when the sun is only partially obscured. A radio guider does not exist.]
2.1.2
Move off-center to locate quiet-sun-center or bursts. Track the quiet sun at a limb, rather than oncenter. When a locatable burst occurs, snap to a point one solar radius away, losing one block of data
during the movement. [But this is complicated, and other difficulties will probably result.]
2.1.3
Dynamically re-center using quiet-sun pointing offsets. Write a file which tracks quiet-sun
pointing offsets, which might reproduce closely every day except for slow drifts related to changing
seasons. Fit each days’ measurements, avoiding outliers, bad fits, etc., and update alignment parameters in
the antenna steering program. Pointing should improve each day. If a burst occurs, then fit the preceeding
sun-center offsets and extrapolate in time. In real-time, adjust tracking to re-center our pointing based on
quiet-sun locations, but try to avoid jitter.
2.1.4
When a burst occurs, start scanning (guided by modulation fits) to "peak up" on it, improving
signal-to-noise, and using our beamwidth to locate it in parallel with the usual algorithms. [This would
have to happen quickly, and may result in messy spectra, taken during movements.]
2.2
Sample differently.
2.2.1
Integrate longer at each frequency to improve signal-to-noise. [However, it is not clear that
random noise dominates flux resolutions. It would take longer to cycle through all patrol frequencies, so
timing precision will degrade.]
2.2.2
Sample more frequencies. This allows more liberal elimination of poorly measured or badly fit
data. [It would take longer to cycle through all patrol frequencies, so timing precision will degrade.]
2.2.3
Go from one cycle of 120 frequencies to 5 cycles of 24 frequencies, thus gain timing resolution.
[But location accuracy could suffer. It is best to sample all frequencies within each modulation period as
close as possible in time, rather than to take samples as much as 5 seconds apart.]
2.2.4
Sample more densly at the upper end of burst spectra. At present, SRBL tunes to about twice as
many frequencies from 2 to 8 Ghz as above 8 GHz. If there is enough high-frequency spectral power in
typical bursts, then locations are improved by inverting this distribution. [The voting scheme for burst
detection must change. Broad, high frequency RFI would be more detrimental than now.]
2.3
Treat spectra and/or fit differently.
2.3.1
At present, SC is not adjusted for gain drifts during the day.
2.3.2
Smooth SC to eliminate spectral fluctuations that are always added, or ignore S C entirely, since it
should be unmodulated.
2.3.3
Use different fitting algorithms.
2.3.4
Weight the variances or a2+b2 with inverse uncertainties as they are accumulated, before assesing
the best grid point.
2.3.5
Use calibration fitting uncertainties to downplay poorly known parameters. Incorporate
calibration uncertainties into the weights used when fitting locations, and thus avoid imprecise (X,Y) or W.
2.3.6
Fix spectral irregularities.
type of irregularity
possible cause
spikes: isolated bad points
RFI
46
hash:
multiple incoherent nearby points broad RFI
breaks: sudden shifts of many nearby pts electronics, software effects
wiggles: coherent bumps or dips
wind shake, fast gain drifts
drifts: broad distortions
gain drift, sky changes
[But there is no way to approximate the complete underlying spectrum. Even many-order polynomial fits
do not accommodate the range of variations seen, including multiple peaks, jagged gain discontinuities, etc.
Instead, spectra must be treated piecemeal, generally within each period, but how?]
2.3.7
Get burst locations from summed spectra, rather than each record separately. Cumulative spectra
acquire progressively better signal to noise. [But opportunities for refinement are lost; e.g. eliminating bad
records or evaluating locations just during event peaks.]
2.3.8
Do a Fourier analysis of patrol spectra, and strip off components not at frequencies corresponding
to modulations.
Pointing
A short, dense description of a related scheme is presented below. This takes advantage of the
same principles as those used to locate bursts, but now to measure boresite-sun offsets quickly, within one
minute...
Observe quiet-sun signals Sij at four frequencies i and four locations (xj,yj) = (xQ+x'j,yQ+y'j),
where (xQ,yQ) are boresite-sun displacements and (x'j,y'j) are commanded offsets; i.e. Sij = Siexp(−2ij/2W2i)
with Si the unmodulated signal, Wi = /i (with   5) and
2ij = (xj−Xi)2 + (yj−Yi)2 where (Xi,Yi) = p(cos,sin)/i are feed offsets (with p  1),
= (x'j2+y'j2) + [(xQ−Xi)2+(yQ−Yi)2] − 2[x'j(xQ−Xi)+y'j(yQ−Yi)]
= 2W2i(lnSi − lnSij). If the four locations lie along pure (az,el) offsets, then
(x'1,y'1)=( a, 0) => a2 + [(xQ−Xi)2+(yQ−Yi)2] − 2a(xQ−Xi) = 2W2i(lnSi − lnSi1)
(x'2,y'2)=( 0, a) => a2 + [(xQ−Xi)2+(yQ−Yi)2] − 2a(yQ−Yi) = 2W2i(lnSi − lnSi2)
(x'3,y'3)=(−a, 0) => a2 + [(xQ−Xi)2+(yQ−Yi)2] + 2a(xQ−Xi) = 2W2i(lnSi − lnSi3)
(x'4,y'4)=( 0,−a) => a2 + [(xQ−Xi)2+(yQ−Yi)2] + 2a(yQ−Yi) = 2W2i(lnSi − lnSi4); i.e.
(xQ−Xi,yQ−Yi) = W2i(ln(Si1/Si3),ln(Si2/Si4))/2a. If in addition, the four frequencies are 90º apart on one feed
cycle, chosen without regard for initial phase, then
(X1,Y1)=p( c, s)/1 => xQ − pc/1 = 2ln(S11/S13)/2a21 & yQ − ps/1 = 2ln(S12/S14)/2a21
(X2,Y2)=p(−s, c)/2 => xQ + ps/2 = 2ln(S21/S23)/2a22 & yQ − ps/2 = 2ln(S22/S24)/2a22
(X3,Y3)=p(−c,−s)/3 => xQ + pc/3 = 2ln(S31/S33)/2a23 & yQ + ps/3 = 2ln(S32/S34)/2a23
(X4,Y4)=p( s,−c)/4 => xQ − ps/4 = 2ln(S41/S43)/2a24 & yQ + ps/4 = 2ln(S42/S44)/2a24
xQ = 2[ln(S11/S13)/1 + ln(S21/S23)/2 + ln(S31/S33)/3 + ln(S41/S43)/4]/2a(1+2+3+4)
yQ = 2[ln(S12/S14)/1 + ln(S22/S24)/2 + ln(S32/S34)/3 + ln(S42/S44)/4]/2a(1+2+3+4)
This is a sum over i of ln(Sij/Sik), with weights i and overall scale 2/2a. Accuracy may be improved by
weighting each ln(Sij/Sik) with i(Vij/S2ij+Vik/S2ik) instead, where Vij are variances on Sij, and then dividing
by the sum of these weights. Operating around 5 GHz and with a = 1º should give a 4º “reach”. Allowing
4 seconds for slewing between locations and 1 second timing errors before and after 4 second data
acquisitions (i.e. 25 samples per frequency) at each offset, the total time required for the trajectory is 40
seconds. Synchronizing data transfer and analysis takes several more seconds.
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