From the raw timeseries, we compute powerspectra from each of the two antennas, and the cross-spectrum. When normalized, this gives the coherence, and it also gives the phase. More on this below.
The figure on the right shows one example of these spectra, this is a 0.2
second integration starting at 06:46:16.40, in a 27 km range gate centered
at 490 km. The first panel in this figure shows the powerspectra from the
32 and 42 meter antennas, on a logarithmic scale. The spectra have the same
overall shape, the 32 m antenna has a higher noise level and a lower power,
all of which is as expected. We should however point out that on occasions,
the power received on the 32 m antenna is higher. The second panel shows
the coherence, or normalized cross-spectrum, as well as the phase of the
cross-spectrum (the crosses). Because of the normalisation, we have removed
points where either of the powerspectra is below a certain threshold. We
can see that the stronger spectral peak is associated with a high coherence,
and with a region of constant phase, which indicates that this enhanced
scattering is due to a localized structure. The lower peak does not
show similar evidence of a localized structure.
When plotting these quantities for all available ranges, a colourplot like the figure to the right is appropriate. Left to right, top to bottom, the panels are:
Powerspectra have been computed in (mostly) the same way as is done in the ordinary ESR receiver system. The received signal in each channel has been mixed and filtered and lagged products have been formed. ACFs have been extracted using the trapezoidal summation (averaging) rule. Spectra have been formed by symmetrizing each ACF estimate and applying a Hanning window before Fourier transforming. The window helps to reduce "ringing" in the spectral estimate, at the cost of some reduction in spectral resolution.
For cross-spectra, lagged cross-products have been formed in much the same way, except that both positive and negative lags were formed. The trapezoidal summation rule (extended to negative lags) was again used, with increasing number of points summed for lags increasing or decreasing from zero. Again, a Hanning window was applied before Fourier transforming into a cross-spectrum.
The coherence is now just the cross-spectrum normalized to the geometric mean of the powerspectra, on a frequency-by-frequency basis. As the powerspectra tend to zero for higher or lower frequencies, this can cause artificially high coherences. We have therefore discarded all points where either of the powerspectra was below a certain threshold.
The effects obtained using a lag window on the correlation functions, these examples are from the 0.2 s integration starting on 06:46:20.60 UT, for the range gate centered at 733 km. The top two panels show correlation function estimates computed from the measurements, real parts in blue, imaginary parts in red. Broken lines are the original estimates, solid lines are after windowing has been applied. The top left panel shows the ACF estimate obtained with the 42 m antenna, while the top right panel shows the XCF estimate. The bottom left panel shows the resulting powerspectra, using rectangular (broken line) and Hanning (solid line) windows, while the bottom right panel shows the resulting coherence (blue lines) and phase (red markers) for rectangular window (broken lines, crosses) and Hanning window (solid lines, diamonds). (EPS)
The effect of the Hanning window relative to no windowing is
shown in detail for a single range gate in the figure on the right. The
top two panels show ACF and XCF estimates before (broken lines) and after
windowing (solid lines). Without the Hanning window, the powerspectrum
(lower left panel) has oscillations at low and high frequencies, and
through the normalization, these are reflected in the coherence (lower
right panel). Applying the window removes these oscillations, resulting
in a smooth powerspectrum, and coherence and cross-spectrum phase with
fewer artificial features, at the cost of some loss of spectral
resolution, seen as a slight smearing of the peaks in the power
spectrum.
The entire profile for the same 0.2 second integration as the previous two figures, here with no lag windowing. (EPS or PDF)
The figure on the right shows the situation for a whole profile. Some points in the top three range gates at slightly negative and large negative frequencies have even higher coherence than the region of uniform high coherence (600-650 km, -15 kHz). Inspection of the powerspectra shows that this is an effect of a "dip" in one or both of these rather than a peak in the cross-spectrum. The smoothed spectrum in the previous figure does not show high coherences at these points.
The spectral estimation introduces convolutions in frequency space. Using
all lags equally is like using a rectangular lag window, which is equivalent
to convolving the true spectrum with a sinc. Strong peaks therefore "ring"
in frequency, resulting in both "dips" in neighbouring frequencies and
oscillating structures easily visible in all four panels of the
non-windowed plot to the right.
When computing cross-spectra, we can modify the phase of one of the signals from one pulse to the next to look for moving structures. We call this phase rotation. Phase increment has been specified in rad/IPP, where the IPP of the experiment is 6.720 ms. What this translates to in terms of velocities along the baseline is of course range dependent.
The following mpeg animations each show one 0.5 second integration from the interesting period, with the phase increment varying from -0.4 to +0.4 rad/IPP in steps of 0.01 rad/IPP.
Maximum structure size vs. coherence for a structure centered in the beam under ideal assumptions (EPS or PDF)
This is perfectly negligible, and if the Gaussian patterns are justifiable, we can safely conclude that scatterers larger than ~ 1/2 the fringe size will not be visible through the noise.
To the right is a plot which shows the structure size (in km) vs. coherence for altitudes 100, 300, 500 and 700 km, under ideal assumptions (no noise). In general, structures at 3-500 km would have to be smaller than 3-500 m in the middle of the beam for their coherence to be visible above the noise floor at 0.2 second integration. The only thing causing a dependence on range is the widening of the antenna beam and interference pattern with range.
As far as I can tell, all sources of error (save one) will only contribute to decrease our observed coherence, hence overestimating the size of the structures. Not subtracting background adds to the power in the spectra used for normalisation -> lower coherence etc. Only points in the powerspectra becoming artificially low (due to "ringing" or noise) will lead to artificially high coherences. With the Hanning window and threshold on the powerspectra, I believe this problem has been adressed, at least for integration times >= 200 ms. This means that a coherence of, say, 0.6 observed at 500 km can be interpreted as beeing, say, 400m wide, with no noise, or it can be narrower (in x-direction) and affected by one or more of these factors to produce a lower observed coherence.
Next step should be to go through the observations and label coherent structures with their (maximum) size. I suppose a size distribution would be interesting, even with the caveats noted above. Once this is done, we can assume filament (sigmay = sigmax) or sheet-like (sigmay large) structure, and compute what enhancement we're really seeing from the scattering volume. (a filamented structure 1/10 of the fringe size would cover ~ 1/2000 of the beam, so a factor of 100 enhancement in total power is really a factor of 200 000 if all the enhanced power comes from the filament! Or did I overlook something here?
I've also tried to see if size in y-direction matters. It doesn't, except as to offset a y displacement. However, a sheet-like structure with even a small angle with the y-axis will be extended in the x-direction, and have its coherence severely limited. We therefore interpret high coherence as a result of basically cylindrical structures.
At 0.2 second integrations, extracting maximum coherence from each of four regions and then deciding which are the result of a coherent scatterer and which are due to noise, we end up with 79 observations of coherent scattering within one minute of data. To the right are two figures which summarize these observations. The first one shows the occurences of coherent structures against range and physical size. Upshifted structures appear at higher altitudes and tend to be larger than downshifted structures at ranges exceeding 600 km. The dashed line indicates that larger structures (i.e. lower coherences) will not be detected using this technique.
The second figure shows, for the cases where simultaneous up- and down-shifted coherent regions were seen, their relative size. I think there are too few occurences for this figure to be really meaningful.
The next two figures show occurence histograms of all observed structure sizes. As the technique cut-off varies with height, we should be careful when interpreting this figure. The first figure shows all observations, the second one have them separated into up- and downshifted observations.
