4.4 Varimax Rotation
4.4.2 NSY, 45
Figures 4.45 to 4.48 show the next four principal components, for comparison with the pre- rotated components in Figures 4.13 to 4.15. PC 3 shows a semi-annual variation in the correla- tions, with peaks around the equinoxes, the largest of these being in March. Comparison of this component with Figure 4.8(d) shows a remarkable similarity between these two curves. Thus, it represents the signal strength variation during autumn and spring.
dB
0 4 8 12 16 20 24
−2−101
residual
0 4 8 12 16 20 24
−0.4−0.20.00.20.4
Figure 4.32: Signal strength of GBZ on 13 December 2007, with the dashed lines on the bottom panel showing times of solar flares with integrated flux>2×10−4 Jm−2.
09:35 09:38 09:41 09:44 09:47 09:50 09:53 09:56 09:59 10:02 10:05 10:08 10:11 10:14
0e+001e−062e−063e−064e−065e−06
Time (UT)
X−ray Flux (W/m^2)
Figure 4.33: X-ray flux for the C4.5 flare with onset at 9:39 UT on 13 December 2007.
dB
0 4 8 12 16 20 24
−2−101
residual
0 4 8 12 16 20 24
−0.4−0.20.00.20.4
Figure 4.34: Signal strength of GBZ on 18 December 2007, showing the times of solar flares with integrated flux>2×10−4 Jm−2.
dB
0 4 8 12 16 20 24
−2.0−1.00.00.51.01.5
residual
0 4 8 12 16 20 24
−0.6−0.4−0.20.00.20.4
Figure 4.35: Signal strength of NSY on 2 November 2008, with times of all solar flares indicated.
dB
0 4 8 12 16 20 24
−3−2−101
residual
0 4 8 12 16 20 24
−0.6−0.4−0.20.00.20.40.6
Figure 4.36: Signal strength of NSY on 3 November 2008, with times of solar flares indicated.
UT
dB
0 4 8 12 16 20 24
−1.5−1.0−0.50.00.51.01.52.0
(a) PC 1
Month/Year
r
06/07 08/07 10/07 12/07 02/08 04/08 06/08 08/08 10/08 12/08 02/09 04/09
−0.4−0.20.00.20.40.60.81.0
(b) Loadings of PC 1
Figure 4.37: First principal component and loadings for GBZ.
UT
dB
0 4 8 12 16 20 24
−2−1012
(a) PC 2
Month/Year
r
06/07 08/07 10/07 12/07 02/08 04/08 06/08 08/08 10/08 12/08 02/09 04/09
−0.20.00.20.40.60.81.0
(b) Loadings of PC 2
Figure 4.38: Second principal component and loadings for GBZ.
UT
dB
0 4 8 12 16 20 24
−2−1012
(a) PC 3
Month/Year
r
06/07 08/07 10/07 12/07 02/08 04/08 06/08 08/08 10/08 12/08 02/09 04/09
−0.8−0.6−0.4−0.20.00.2
(b) Loadings of PC 3
Figure 4.39: Third principal component and loadings for GBZ.
UT
dB
0 4 8 12 16 20 24
−2−1012
(a) PC 4
Month/Year
r
06/07 08/07 10/07 12/07 02/08 04/08 06/08 08/08 10/08 12/08 02/09 04/09
−0.6−0.4−0.20.00.20.40.6
(b) Loadings of PC 4
Figure 4.40: Fourth principal component and loadings for GBZ.
UT
dB
0 4 8 12 16 20 24
−4−3−2−1012
(a) PC 5
Month/Year
r
06/07 08/07 10/07 12/07 02/08 04/08 06/08 08/08 10/08 12/08 02/09 04/09
−0.20.00.20.40.60.8
(b) Loadings of PC 5
Figure 4.41: Fifth principal component and loadings for GBZ.
UT
dB
0 4 8 12 16 20 24
−2−1012
(a) PC 6
Month/Year
r
06/07 08/07 10/07 12/07 02/08 04/08 06/08 08/08 10/08 12/08 02/09 04/09
−0.6−0.4−0.20.00.20.4
(b) Loadings of PC 6
Figure 4.42: Sixth principal component and loadings for GBZ.
UT
dB
0 4 8 12 16 20 24
−1.5−1.0−0.50.00.51.01.5
(a) PC 1
Month/Year
r
07/08 08/08 09/08 10/08 11/08 12/08 01/09 02/09 03/09 04/09 05/09
0.00.20.40.60.8
(b) Loadings of PC 1
Figure 4.43: First principal component and loadings for NSY.
UT
dB
0 4 8 12 16 20 24
−4−3−2−101
(a) PC 2
Month/Year
r
07/08 08/08 09/08 10/08 11/08 12/08 01/09 02/09 03/09 04/09 05/09
0.00.20.40.60.8
(b) Loadings for PC 2
Figure 4.44: Second principal component and loadings for NSY.
UT
dB
0 4 8 12 16 20 24
−3−2−1012
(a) PC 3
Month/Year
r
07/08 08/08 09/08 10/08 11/08 12/08 01/09 02/09 03/09 04/09 05/09
0.00.20.40.60.8
(b) Loadings of PC 3
Figure 4.45: Third principal component and loadings for NSY.
UT
dB
0 4 8 12 16 20 24
−101234
(a) PC 4
Month/Year
r
07/08 08/08 09/08 10/08 11/08 12/08 01/09 02/09 03/09 04/09 05/09
−0.8−0.6−0.4−0.20.00.20.4
(b) Loadings of PC 4
Figure 4.46: Fourth principal component and loadings for NSY.
UT
dB
0 4 8 12 16 20 24
−2−1012
(a) PC 5
Month/Year
r
07/08 08/08 09/08 10/08 11/08 12/08 01/09 02/09 03/09 04/09 05/09
−0.20.00.20.4
(b) Loadings of PC 5
Figure 4.47: Fifth principal component and loadings for NSY.
UT
dB
0 4 8 12 16 20 24
−4−202468
(a) PC 6
Month/Year
r
07/08 08/08 09/08 10/08 11/08 12/08 01/09 02/09 03/09 04/09 05/09
−0.6−0.4−0.20.00.2
(b) Loadings of PC 6
Figure 4.48: Sixth principal components and loadings for NSY.
PC 4 has a mostly small, negative correlation with the majority of variables, with the loadings becoming increasingly negative towards the summer. This component acts to level out the daytime signal of the summer months, also moving the dawn and dusk terminator dips to earlier and later times respectively.
The generally random distributions of loadings on PC 4 for GBZ is seen here in PC 5. This component also acts to add an asymmetrical bias between the pre-dawn and post-dusk nighttime signal strength. The sixth principal component is only strongly anti-correlated in September with a few variables and adds a strong oscillation to the signal at around 04:30 UT.
Although the analysis of the effect of each principal component is still complicated, the varimax rotation has made it simpler to determine the time of year when the component has the most influence on the signal variation. This is due to the fact that the loadings are either strong or weak and no longer strongly positive and strongly negative and hence each variable is defined by only a few strong correlations with the principal components.
Chapter 5
Conclusions
Subionospherically propagating VLF signals are strongly influenced by conditions in the iono- sphere. Diurnal and annual variations arise in the signals due to the corresponding variations in the ionosphere. As there are many factors which influence the bottom of the ionosphere, on a periodic and transient basis, the consequent variations in the signals are quite complex.
Narrowband VLF data have been analysed using a multivariate data analysis technique, prin- cipal components analysis, in an attempt to unravel some of these complex variations. Once the standard annual and diurnal variation is known, transient events can be detected in the deviations of the signal from the expected norm.
There were two transmitters used in this analysis. The data for GBZ, operating at 19.6 kHz, covered a span of almost two years, while the data from NSY, operating at 45.9 kHz, was just under a year long. This allowed for some interesting analysis of both the diurnal and annual variations in the signals.
The days on which operational problems with the transmitter or receiver caused the recording of bad data were neglected when the PCA was performed. This is due to the common sense rationale that goes with any computational analysis: garbage in equals garbage out. The gaps in the data time series caused by removing the bad data did not adversely affect the results of the PCA. In fact, the results of the PCA may be used to construct an estimation of the quiet signal amplitude variation on missing days, either by a crude average of the surrounding days, or by an interpolation between the reconstructed signals.
Quiet time estimation of the signals, acquired by reconstructing the variables without the higher order components, performed consistently better than a Fourier analysis. This is because PCA reconstruction uses basis functions that are derived from the data itself, while the Fourier analysis is based on generic mathematical functions.
PCA for GBZ indicated the first nine principal components collectively accounted for 95%
of the variance in the data. By summing the relevant amounts of each of these components an approximation of the signal for any day within the data set can be created. Examination of each principal component and its component loadings also yields valuable information. The first principal component is one on which all the variables are strongly positively correlated. This shows the general diurnal variation seen throughout the year. The higher order components then add effects typical of each season, moving the dawn and dusk terminators, changing their shape, and changing the signal level of both the daytime and nighttime transmission. The times at which each component is most important is seen in high correlation values of their loadings.
Although the PCA of NSY did not perform quite as well as that for GBZ, similar results were obtained. NSY, being a north-south signal, varies more drastically over the course of a year than GBZ. This is due to the differing times at which the terminator crosses the signal path and the differing angle at which this occurs throughout the year. During the equinoxes, the terminator line would be almost parallel to the signal path, meaning that the change between day and night
occurs rapidly. This greater annual variability of NSY is reflected in the number of components required to account for 95% of the variance in the data, being 28 components.
Both the GBZ–Tihany and NSY–Tihany paths are relatively short (less than 2000 km). These results would be interesting in comparison to the equivalent from a longer signal path.
The residuals between the reconstructed curve and the original data, of both transmitters, were examined for the effects of transient events. These transient events are expected to cause additional ionisation in the ionosphere, causing a modification in the Earth-ionosphere waveguide, thereby changing the propagation conditions of the signals.
The residuals were examined for the effects of over 390 GRBs occurring over the course of 2007, 2008 and the first half of 2009. No effects were seen in either the residuals, or the raw data, for either transmitter. The observation of a transient effect in the VLF data caused by a GRB requires the following things:
• a long burst, with duration exceeding 100 s;
• the fluence of the burst must be greater than ∼10−7 Jm−2;
• the sub-burst point must be in reasonable proximity to the transmitter-receiver paths;
• the signal path should be in darkness;
• and both the transmitter and receiver need to be operational.
A confluence of these factors did not occur within the time frame of the data.
The 145 TGFs occurring during the time frame of the data set also did not create an effect in the residuals. Although there were some alignments of fluctuations with the times of the TGFs in the residuals, these flashes occurred too far away to conceivably have an effect. As the TGF occurs within the neutral atmosphere, and γ-rays are absorbed by this medium, the TGF signal would not be detected over the horizon and hence a VLF-perturbing TGF would need to occur over the transmitter path.
Numerous solar flare perturbations were detected in the residual data. The lower total inte- grated flux limit for detection of an event on GBZ was determined as 2×10−4Jm−2 and that for NSY was 7.5×10−5 Jm−2. Previous events detected in VLF data put the lowest class of the event that could cause an ionospheric effect as a C1 flare. However, the residuals method is much more sensitive to perturbations and thus lower peak flux events are detected in them. The lowest class event that was detected was a B2.4 flare, which has considerably lower peak flux than a C1 flare. However, as flares with a higher class and total integrated flux below the defined limits were not detected, the total integrated flux seems to be a more important factor in the occurrence of a VLF perturbation than the peak flux.
Some of the higher peak flux flares were also detected in the raw data, however, the signature in the residuals is greatly accentuated. The perturbations generally last slightly longer than the duration of the flare, with the onset time not delayed appreciably. Both increases and decreases in amplitude of the signals are seen to be caused by the flares. There has also been detection of flare perturbation occurring when the signal paths are in darkness. The times of these flares, in terms of onset, peak and end, are too well correlated with the perturbations to claim they are chance alignments.
Further work was done on the PCA components, rotating them to a new, more interpretable orientation via a varimax rotation. This rotation searches for simple structure in the component loadings matrix, thereby ensuring that each component is only well correlated with a few variables and each variable is defined by only a few components.
The results of the varimax rotation yielded very clear summer, winter and equinox signals.
There was also a component that, while not vitally important to any of the variables, was either
positively or negatively correlated with most of them. This method has potential in identifying longer term variations in the amplitude of subionospherically propagating signals, such as might be caused by the eleven year solar cycle, given sufficient data.
Overall, PCA has proven to be a powerful analysis tool, being used both in the detection of transient events and the development of a quiet day approximation to the narrowband, subiono- spherically propagating data.
There are many aspects of this research that would benefit from further analysis. PCA applied to signal paths of much longer length would provide for an interesting comparison of the signal variations between different paths. Examination of data covering a larger period of time would allow longer duration variations in the signals to be identified.
The search for perturbations caused by TGFs could be repeated on a signal path that crossed the regions of highest density of TGF occurrence, thus clarifying the possibility of these events causing a perturbation.
Lastly, a more extensive examination of solar flare perturbations, in relation to total integrated flux, peak flux, time of day and time of year may determine the cause of the perturbations detected when the signal paths are in darkness. This would also allow the conditions for a flare to cause a perturbation to be more well defined.
Bibliography
T. Bai and P. A. Sturrock. Classification of Solar Flares. Annual review of astronomy and astrophysics, 27:421–467, 1989.
W. C. Bain and E. Hammond. Ionospheric solar flare effect observations. Journal of Atmospheric and Terrestrial Physics, 37:573–574, 1975.
D. Band, J. Matteson, L. Ford, B. Schaefer, D. Palmer, B. Teegarden, T. Cline, M. Briggs, W. Paciesas, G. Pendleton, G. Fishman, C. Kouveliotou, C. Meegan, R. Wilson, and P. Lestrade.
BATSE observations of gamma-ray burst spectra. I - Spectral diversity. The Astrophysical Journal, 413:281–292, Aug. 1993. doi: 10.1086/172995.
R. Barr, D. Llanwyn Jones, and C. J. Rodger. ELF and VLF radio waves. Journal of Atmospheric and Terrestrial Physics, 62:1689–1718, 2000.
R. T. Brown. Ionospheric effects of cosmic gamma-ray bursts. Nature, 246:83–84, 1973. doi:
10.1038/246083a0.
K. G. Budden. The ”Waveguide Mode” Theory of the Propagation of Very-Low-Frequency Radio Waves. Proceedings of the IRE, 36:772–774, 1957.
B. W. Carroll and D. A. Ostlie. An Introduction to Modern Astrophysics. Addison-Wesley Pub- lishing Company, Inc., 1996.
M. W. Chevalier and U. S. Inan. A Technique for Efficiently Modeling Long-Path Propagation for Use in Both FDFD and FDTD. IEEE Antennas and Wireless Propagation Letters, 5:525–528, 2006.
M. A. Clilverd, N. R. Thomson, and C. J. Rodger. Sunrise effects on VLF signals propagating over a long north-south path. Radio Science, 34:939–948, 1999.
D. D. Crombie. Periodic fading of VLF signals received over long paths during sunrise and sunset.
Journal of Research National Bureau of Standards, Radio Science, 68D:27–34, 1964.
S. A. Cummer. Modeling Electromagnetic Propagation in the Earth-Ionosphere Waveguide. IEEE Transactions on Antennas and Propagation, 48:1420–1430, 2000.
J. Cummings, S. D. Barthelmy, E. Fenimore, N. Gehrels, H. Krimm, C. Markwardt, K. McLean, D. Palmer, A. Parsons, J. L. Racusin, T. Sakamoto, G. Sato, M. Stamatikos, J. Tueller, and T. Ukwatta. GRB 080319B, Swift-BAT refined analysis. GCN Circular 7462, 2008.
J. R. Dwyer. Source mechanisms of terrestrial gamma-ray flashes.Journal of Geophysical Research, 113, D10103, 2008. doi: 10.1029/2007JD009248.
J. R. Dwyer and D. M. Smith. A comparison between Monte Carlo simulations of runaway breakdown and terrestrial gamma-ray flash observations. Geophysical Research Letters, 32, L22804, 2005. doi: 10.1029/2005GL023848.
C.-F. Enell, P. T. Verronen, M. J. Beharrell, J. P. Vierinen, A. Kero, A. Sepala, F. Honary, T. Ulich, and E. Turunen. Case study of the mesospheric and lower thermospheric effects of solar X-ray flares: Coupled ion-neutral modelling and comparison with EISCAT and riometer measurements. Annales Geophysicae, 26:2311–2321, 2008.
E. E. Fenimore. The Average Temporal and Spectral Evolution of Gamma-Ray Bursts. The Astrophysical Journal, 518:375–379, June 1999. doi: 10.1086/307245.
G. J. Fishman and U. S. Inan. Observation of an ionospheric disturbance by a gamma-ray burst.
Nature, 331:418–420, 1988.
N. Gehrels, G. Chincarini, P. Giommi, K. O. Mason, J. A. Nousek, A. A. Wells, N. E. White, S. D. Barthelmy, P. W. A. Roming, L. Angelini, D. N. Burrows, L. R. Cominsky, K. C. Hurley, F. E. Marshall, P. Me, L. M. Barbier, T. Belloni, S. Campana, P. A. Caraveo, M. M. Chester, O. Citterio, T. L. Cline, M. S. Cropper, J. R. Cummings, A. J. Dean, E. D. Feigelson, E. E.
Fenimore, D. A. Frail, A. S. Fruchter, G. P. Garmire, K. Gendreau, G. Ghisellini, J. Greiner, J. E. Hill, S. D. Hunsberger, H. A. Krimm, S. R. Kulkarni, P. Kumar, F. Lebrun, C. B.
Markwardt, B. J. Mattson, R. F. Mushotzky, J. P. Norris, J. Osborne, B. Paczynski, D. M.
Palmer, A. M. Parsons, J. Paul, M. J. Rees, C. S. Reynolds, J. E. Rhoads, T. P. Sasseen, B. E. Schaefer, A. T. Short, A. P. Smale, I. A. Smith, L. Stella, G. Tagliaferri, T. Takahashi, M. Tashiro, L. K. Townsley, J. Tueller, M. J. L. Turner, M. Vietri, W. Voges, M. J. Ward, R. Willingale, F. M. Zerbi, and W. W. Zhang. The swift gamma-ray burst mission. The Astrophysical Journal, 611:1005–1020, 2004.
S. Golenetskii, R. Aptekar, E. Mazets, V. Pal’shin, D. Frederiks, and T. Cline. Konus-Wind observation of GRB 080319B. GCN Circular 7482, 2008.
P. E. Green and J. D. Carroll. Analyzing Multivariate Data. The Dreyden Press, 1978.
B. W. Grefenstette, D. M. Smith, B. J. Hazelton, and L. I. Lopez. First RHESSI terrestrial gamma ray flash catalog. Journal of Geophysical Research, 114, 2009. doi: 10.1029/2008JA013721.
D. P. Grubor, D. M. Sulic, and V. Zigman. Classification of X-ray solar flares regarding their effects on the lower ionosphere electron density profile. Annales Geophysicae, 26:1731–1740, 2008.
A. V. Gurevich, G. M. Milikh, and R. Roussel-Dupre. Runaway electron mechanism of air break- down and preconditioning during a thunderstorm. Physics Letters A, 165:463–468, 1992. doi:
10.1016/0375-9601(92)90348-P.
B. J. Hazelton, B. W. Grefenstette, D. M. Smith, J. R. Dwyer, X.-M. Shao, S. A. Cummer, T. Chronis, E. H. Lay, and R. H. Holzworth. Spectral dependence of terrestrial gamma-ray flashes on source distance. Geophysical Research Letters, 36, L01108, 2009. doi: 10.1029/
2008GL035906.
U. S. Inan, A. Slingeland, and V. P. Pasko. VLF and LF signatures of mesospheric/lower iono- spheric response to lightning discharges. Journal of Geophysical Research, 101:5219–5238, 1996.
U. S. Inan, N. G. Lehtinen, S. J. Lev-Tov, M. P. Johnson, T. F. Bell, and K. Hurley. Ionization of the lower ionosphere by gamma-rays from a magnetar: detection of a low energy (3 – 10 keV) component. Geophysical Research Letters, 26:3357–3360, 1999.
K. Kasturirangan, U. R. Rao, D. P. Sharma, R. G. Rastogi, and S. C. Chakravarty. Ionospheric effects of transient celestial x-ray and gamma-ray events. Astrophysics and Space Science, 42:
57–62, 1976. doi: 10.1007/BF00645524.
K. J. W. Lynn. Anomalous sunrise effects observed on a long transequatorial VLF Propagation path. Radio Science, 2:521–530, 1967.
W. M. McRae and N. R. Thomson. Solar flare induced ionospheric D-region enhancements from VLF phase and amplitude observations. Journal of Atmospheric and Terrestrial Physics, 66:
77–87, 2004. doi: 10.1016/j.jastp.2003.09.009.
P. I. Meszaros. Gamma-ray bursts. Reports on Progress in Physics, 69:2259–2321, 2006. doi:
10.1088/0034-4885/69/8/R01.
R. J. Nemiroff, J. W. Bonnell, and J. P. Norris. Temporal and spectral characteristics of terrestrial gamma flashes. Journal of Geophysical Research, 102:9659–9665, 1997.
J. P. Norris, R. J. Nemiroff, J. T. Bonnell, J. D. Scargle, C. Kouveliotou, W. S. Paciesas, C. A.
Meegan, and G. J. Fishman. Attributes of pulses in long bright gamma-ray bursts. The Astrophysical Journal, 459:393–412, 1996.
G. S. Paiva, A. C. Pavao, and C. C. Bastos. Seed electrons from muon decay for runaway mechanism in the terrestrial gamma ray flash production. Journal of Geophysical Research, 114, D03205, 2009. doi: 10.1029/2008JD010468.
R. A. Pappert and D. G. Morfitt. Theoretical and experimental sunrise mode conversion results at VLF. Radio Science, 10:537–546, 1975.
T. Piran. The physics of gamma-ray bursts. Reviews of Modern Physics, 76:1143–1210, Jan. 2005.
doi: 10.1103/RevModPhys.76.1143.
G. L. Pollack and D. R. Stump. Electromagnetism. Pearson Education, Inc., 2002.
R. D. Preece, M. S. Briggs, R. S. Mallozzi, G. N. Pendleton, W. S. Paciesas, and D. L. Band. The BATSE Gamma-Ray Burst Spectral Catalog. I. High Time Resolution Spectroscopy of Bright Bursts Using High Energy Resolution Data. The Astrophysical Journal Supplement Series, 126:
19–36, Jan. 2000. doi: 10.1086/313289.
J. L. Racusin, S. V. Karpov, M. Sokolowski, J. Granot, X. F. Wu, V. Pal’shin, S. Covino, A. J.
van der Horst, S. R. Oates, P. Schady, R. J. Smith, J. Cummings, R. L. C. Starling, L. W.
Piotrowski, B. Zhang, P. A. Evans, S. T. Holland, K. Malek, M. T. Page, L. Vetere, R. Margutti, C. Guidorzi, A. P. Kamble, P. A. Curran, A. Beardmore, C. Kouveliotou, L. Mankiewicz, A. Melandri, P. T. O’Brien, K. L. Page, T. Piran, N. R. Tanvir, G. Wrochna, R. L. Aptekar, S. Barthelmy, C. Bartolini, G. M. Beskin, S. Bondar, M. Bremer, S. Campana, A. Castro- Tirado, A. Cucchiara, M. Cwiok, P. D’Avanzo, V. D/’Elia, M. D. Valle, A. de Ugarte Postigo, W. Dominik, A. Falcone, F. Fiore, D. B. Fox, D. D. Frederiks, A. S. Fruchter, D. Fugazza, M. A. Garrett, N. Gehrels, S. Golenetskii, A. Gomboc, J. Gorosabel, G. Greco, A. Guarnieri, S. Immler, M. Jelinek, G. Kasprowicz, V. La Parola, A. J. Levan, V. Mangano, E. P. Mazets, E. Molinari, A. Moretti, K. Nawrocki, P. P. Oleynik, J. P. Osborne, C. Pagani, S. B. Pandey, Z. Paragi, M. Perri, A. Piccioni, E. Ramirez-Ruiz, P. W. A. Roming, I. A. Steele, R. G. Strom, V. Testa, G. Tosti, M. V. Ulanov, K. Wiersema, R. A. M. J. Wijers, J. M. Winters, A. F.
Zarnecki, F. Zerbi, P. M´esz´aros, G. Chincarini, and D. N. Burrows. Broadband observations of the naked-eyeγ-ray burst GRB 080319B. Nature, 455:183–188, 2008. doi: 10.1038/nature07270.
Y. Reuveni and C. Price. A new approach for monitoring the 27-day solar rotation using VLF radio signals on the Earths surface. Journal of Geophysical Research, 114, A10306, 2009. doi:
10.1029/2009JA014364.