Real-Time Ionospheric Scintillation Monitoring

Real-Time Ionospheric Scintillation Monitoring Wanxuan Fu, Shaowei Han and Chris Rizos The University of New South Wales, Australia Mark Knight and Anthony Finn Defence Science & Technology Organisation, Australia BIOGRAPHY Dr. Wan Xuan Fu joined the School of Geomatic Engineering, The University of New South Wales, Australia, in 1989, as PhD student first, and later as a Senior Research Associate. His research interests are in integrated navigation system, signal processing and system control. Shaowei Han is a Senior Lecturer in the School of Geomatic Engineering, The University of New South Wales (UNSW), Australia. His research interests are in GPS/GLONASS ambiguity resolution and error mitigation methods for real-time carrier phase-based kinematic positioning over short-, medium-, and longrange, GPS attitude determination and the integration of GPS, INS and Pseudolite. He is Chairman of International Association of Geodesy Special Study Group 1.179 “ Wide (regional) area modelling for precise satellite positioning”, and has authored over 100 journal and conference publications. Chris Rizos is an Associate Professor at the School of Geomatic Engineering, UNSW, where he is leader of the Satellite Navigation and Positioning (SNAP) group. SNAP is the premier academic GPS R&D group in Australia, specializing on the technology for precise static and kinematic applications of GPS. Mark Knight is an engineer with Surveillance Systems Division (SSD) of the Defence Science & Technology Organisation (DSTO) in Adelaide, South Australia. Since graduating from the University of Adelaide in 1987 he has worked in the fields of microwave radar and GPS navigation systems and is currently working towards a PhD degree in Engineering science. His principal area of interest is in the effects of the ionosphere on GPS performance. Anthony Finn is a senior research scientist with SSD and heads the Satellite Navigation section. Since graduating from Cambridge University with a PhD he has worked in the field of HF radio wave propagation within the ionosphere and navigation systems. His interests include the effects of ionospheric activity and electromagnetic performance. interference on GPS systems ABSTRACT As the millennium approaches we are faced with a period of increasing solar flux and intense ionospheric scintillation. Not only single-frequency, but also dualfrequency GPS measurements will be badly affected by deep scintillation, due to its temporal and spatial decorrelation characteristics. Using the scintillation observables derived from dual-frequency phase and signal-to-noise ratio measurements, real-time scintillation detection, tracking and indication algorithms have been developed. By an orthonormal projection of the scintillation observables to the timefrequency spaces of different scales, and using multiscale statistical hypothesis tesing, a fast and reliable scintillation detection and tracking can be achieved. A fuzzy expression for “scintillation depth” is also developed to overcome the ambiguity existing in the numerical and linguistic definitions of scintillation intensity. 1. INTRODUCTION Ionospheric scintillation of a radio signal is a relatively rapid fluctuation of the amplitude, phase and Faraday rotation angle of the signal about mean levels which are either constant, or changing much more slowly than the scintillations themselves. Ionospheric scintillation is caused by irregularities of the ionospheric electron density along the signal propagation path. The percent fluctuations of the irregularities are usually very small, but can be as large as nearly 100% at the equator [1]. The irregularities are predominantly in the F-layer of the ionosphere at altitudes ranging from 200 to 1000km with the primary disturbance region being typically between 250 and 400km. Signal fluctuation due to ionospheric irregularities in the F-layer have been reported at frequencies as high as 7GHz [2]. There are times when E-layer irregularities in the 90 to 100km region produce scintillation, particularly sporadic E and auroral E, but they have little effect on L-band GPS signals. 5) The worst source of scintillation is at the equatorial anomaly region, which corresponds to two belts, each several degrees wide, of enhanced ionization in the Flayer at approximately 15o N and 15o S of the magnetic equator. In this region, during the solar cycle maxima periods, amplitude fading at 1.5GHz may exceed 20dB for several hours after sunset [3]. The other potential active scintillation regions are at auroral and polar cap latitudes. In the central polar cap during years of solar cycle maximum, GPS receivers may suffer >10dB fade. The boundary of high and midlatitude region is between 45o and 55o CGL (Corrected Geomagnetic Latitude), where weak irregularities usually commence. The irregularities in the polar region are strongly correlated with solar flux. Even with low magnetic activity, during a year of high solar flux a dramatic increase in the intensity and occurrence of scintillation will experienced. Polar region scintillation activity is a relatively poorly investigated phenomenon. What takes place at middle-latitudes is an extension of phenomena at equatorial and auroral latitudes. The perturbing effects of those regions, and the higher electron densities during high sunspot number years or magnetic storm periods, might combine to provide effects along the line of force thus extending the irregularity boundaries towards the middle-latitude regions. Real-time ionospheric scintillation monitoring (RTISM) is expected to fulfil the following tasks: 1) Scintillation occurrence detection. 2) Scintillation tracking. 3) Scintillation intensity indication. The scintillation observables are derived from dualfrequency GPS phase and signal-to-noise ratio measurements. As ionospheric scintillation has different characteristics from the background ionosphere in both the time and frequency domains, the signal feature can be detected or extracted effectively by transforming the time domain scintillation signals into time-frequency domain signals. The transform is carried out using orthonormal and compactly-supported wavelet bases with the following characteristics: 1) By projection of the signal to the spaces spanned by the wavelet bases at different scales, the signal feature can be localized in both the time and frequency domains. 2) High resolution for both low and high frequency components can be achieved by the automatic zooming in and out of the wavelet basis function [4]. 3) Low-order and compactly-supported wavelet bases can track the signal feature with minimum time delay. 4) As ionospheric scintillations usually have a rapidly changing signature at their beginning, low-order 6) 7) 8) wavelet bases with less regularity can effectively detect the scintillation front effectively. Wavelet transform is linear, hence the Gaussian property is preserved after transformation. The transform is orthogonal, hence no correlation is introduced. The wavelet transform can easily deal with the non-stationary process; hence no pre-filtering of the scintillation signal is needed. The algorithms of the transform are simple, and computations are faster than for the FFT [5]. Both scintillation detection and tracking are binary statistical decision problems which can be solved using Neyman-Pearson detectors, which can achieve maximum detection power for a specific false alarm level with less a priori information. For wavelettransformed scintillation signals, a multi-layer NeymanPearson detector has been developed to increase the efficiency and reliability of scintillation detection. The traditional scintillation depth indicator is based on the normalized standard deviation of the intensity scintillation S 4 and the standard deviation of the phase scintillation σ δΦ . The descriptive indicators of scintillation depth, such as ‘strong’, ‘medium’ and ‘weak’ scintillation, are derived or related to the prescriptive values of S 4 and σ δΦ , but with a certain linguistic ambiguity or imprecision. The main source of this “fuzziness” is the imprecision involved in the definition and use of symbols. For example, S 4 ≥ 0.7 means a ‘strong’ scintillation according to the traditional definition. However, what about S 4 = 0.69999 , does it belong to the ‘strong’ or ‘medium’ set? Under crisp set and Boolean logic, S 4 = 0.69999 definitely belongs to the ‘medium’ set, if the boundary between ‘strong’ and ‘medium’ sets are defined as S 4 = 0.7 . In fact there is no “crisp” boundary between ‘strong’ and ‘medium’ scintillation, and the change from ‘medium’ to ‘strong’ is gradual, not abrupt or sudden, and hence S 4 = 0.69999 should belong “more” to the ‘medium’ set than the ‘strong’ set. The same also applies to the boundary between ‘weak’ and ‘medium’ scintillation. A fuzzy indication system was therefore developed to deal with this problem. ‘Phase scintillation’, or ‘amplitude scintillation’, are defined as linguistic variables, whose arguments, such as ‘strong’, ‘medium’ and ‘weak’, are fuzzy numbers (modeled by fuzzy sets). The universe discourse of the fuzzy sets is the value of S 4 or σ ∆Φ , and the membership functions map every element of the universe of discourse to the interval [0,1]. Since the interval [0,1] contains an infinity of numbers, infinite degrees of membership are possible, which corresponds to a gradation of the definition. 2. SCINTILLATION STATISTICS Some scintillation statistics concerning our experiments are listed below: 1) The electron density irregularity of ionosphere ∆N is supposed to be a zero-mean Gaussian process, its power spectrum density (PSD) following powerlaw rules. Generally the process is non-stationary under the scintillation condition [8]. 2) Phase scintillation ∂Φ is supposed to be a linear function of ∆N , hence it is a Gaussian process [9]. 3) Amplitude scintillation ∂A is not a linear function of ∆N due to the Fresnel filtering and saturation effects. The amplitude scintillation model accepted generally is the Nakagami-m distribution for weak and moderate scintillation, and the Rayleigh distribution for strong scintillation [10]. 4) Although ∂A and ∂Φ are both proportional to ∆N , the correlation between ∂A and ∂Φ is not very clear and no analytic solution is available. Actually under the assumption of Nakagami-m and Rayleigh distributions, ∂A and ∂Φ are assumed to be independent and ∂Φ follows a uniform distribution. 5) Amplitude scintillation index S4 [11]: S4 = 6) 7) 8) 9) (< I 2 ) > − < I >2 / < I >2 signals. As narrow-band signals (bandwidth less than 1Hz), ionospheric scintillation will pass through almost all kinds of unaided DLL (Delay-Lock-Loop) and PLL (Phase-Lock-Loop), and have the potential to affect both single- and dual-frequency receivers, and the performance of the SPS and PPS. The main impacts of ionospheric scintillation on GPS can be summarized as: 1) Amplitude scintillation is modeled as multiple noise [13]: Ar = A0 * ∂A where Ar is the amplitude of the received signal, and A0 is the nominal amplitude of the signal. Phase scintillation is modeled as additive noise: Φr = Φ0 + ∂Φ 2) (1) where I is the signal intensity. Generally S4≥0.7 is considered strong scintillation. Due to the saturation effects, S4 is rarely larger than 1.0, except under conditions of very strong scintillation. Phase scintillation index is the standard deviation of signal phase δ∂Φ, and δ∂Φ≥1.0 rad. is considered to indicate strong scintillation. A frequency dependency of scintillation f −1.5 gives reasonable results for gigahertz signals. Both amplitude and phase scintillation indicate a power-law PSD of f − n . For high frequency, the value of n is between 2 to 4 and is usually not an integer, suggesting that scintillations are chaotic systems with a system order between 1 and 2. There is an apparent cut-off frequency (Fresnel frequency) for the PSD of amplitude scintillation due to the Fresnel filtering and saturation effects. Typically the cut-off frequency is less than 1Hz, but it can reach up to 3Hz under strong scintillation [12]. For phase scintillation no apparent cut-off frequency appears, and its main power is generally distributed in a range of less than 1Hz. 3) 4) 5) 6) 3. SCINTILLATION IMPACTS ON GPS L-band GPS signals are mainly affected by the F-layer irregularities of the ionosphere. As most GPS receivers use omni-directional and circular-polarized antennae, only amplitude and phase scintillation appears in GPS (2) 7) (3) where Φr is the phase measurement, Φ0 is the nominal phase value and ∂Φ is the phase scintillation. ∂A affects directly the C/N0 (signal-to-noise ratio) of the received signals and the measurement noise of both code and phase measurements. The nominal C/N0 for the L1 signal is about 45 dB-Hz, and tracking may be interrupted when C/No is less than 24 dB-Hz. It will be worse for the L2 signal because its power is 6dB lower than that of L1.. Because the tracking error variance is proportional to the bandwidth divided by C/No [14], narrowband tracking loops will be less affected by amplitude scintillation. As a narrow-band process, scintillation can be considered as a modulating signal on the GPS signals. Therefore, the spectrum of received signals will be broadened by a value that is proportional to ∂Φ. From the point of view of satellite signal tracking, PLLs with wider bandwidth work better than the narrow ones. The signal spectrum broadening results in the shortening of the signal correlation time, which is equivalent to an increase in the measurement random noise. This is the temporal decorrelation effect of scintillation on signals. The deeper the scintillation is, the stronger the decorrelation. Under the frozen field assumption (irregularity structure does not change, or changes slowly), the temporal decorrelation is equivalent to spatial decorrelation. The correlation distance of amplitude is generally less than the first Fresnel zone size, which is about 300m for L1 signals [15]. The correlation time is inversely proportional to the irregularity drift velocity and the square root of the signal frequency. Under scintillation conditions, the refraction index of the ionospheric plasma is no longer a linear function of irregularity density, especially when the scintillation is deep and multi-scattering occurs. The propagation path difference of signals with different frequencies cannot be negligible, the usual f −2 relation no longer is valid and frequency decorrelation occurs. For L-band signals an average frequency scaling law f −1.5 gives reasonable results for weak scintillation (S4<0.4) [16]. Actually the power index is not a constant, it varies according to the scintillation depth. 8) As scintillation broadens the signal PSD, the signals become whiter. The lower the frequency is, the more white the signals become, and the stronger the decorrelation of different frequency signals. Even when the third GPS signal is introduced, ionospheric scintillation will still be the main error source for the ionosphere correction to observations made with dual- or triple-frequency receivers when scintillation activity is present 4. REAL-TIME IONOSPHERIC SCINTILLATION MONITORING (RTISM) METHODOLOGY AND ALGORITHMS 4.1 Ionospheric Scintillation Observables For phase scintillation, the observable is [17]: ∆Φ(t ) = Φ L1,l 2 (t ) − Φ L1,l 2 (t − 1) (4) and Φ L1,l 2 (t ) = C[Φ L 2 (t ) − Φ l1 (t )] (5) where Φ L1, ΦL2 are the GPS L1 and L2 phase measurements, and C is a constant. The measurement errors that are independent of signal frequency are eliminated in eqn (5), and constant or low frequency errors are eliminated or suppressed by eqn (4). The remaining error sources are the random errors of multipath, ionosphere, which both are functions of satellite elevation angle, and receiver noise. The random errors are considered to follow the Gaussian distribution. If a realization of the scintillation process is considered to be deterministic signals, then ∆Φ follows the Gaussian distribution in both the noscintillation and with-scintillation cases:  N(mΦ0 , d 2σ 2∆Φ ) no scint.  ∆Φ ∝  2 ) with scint.  N(mΦ1 ,d2 σ ∆Φ (6) where mΦ 0 is the mean of ∆Φ when no scintillation occurs (and is supposed to be zero), mΦ1 is the mean of 2 ∆Φ when scintillation occurs, σ ∆Φ is the variance of measurement noise, and d is a proportional factor, assumed to be inversely proportional to satellite elevation angle. For amplitude scintillation, the observable is: ∆A(t) = C/N0(t) - C/N0(t-1) (7) The reasons we choose the change of C/N0 (or change rate, if ∆C/N0 is normalized by time) are: 1) As indicators of received signal intensity, only the C/N0 values from L1 and L2 tracking loops are available from GPS receivers. The ideal observable for amplitude scintillation would be the input signal power level, or input signal C/N0,, which can be obtained only before the antenna pre-amplifier and limiter. However, because the amplitude scintillation is a narrow-band signal, and well within the linear working range of the front-end RF components, the GPS signal and scintillation signal can be assumed to be amplified proportional to their input power. 2) Usually the received signal amplitude does not follow the Gaussian distribution. For a typical narrow-band modulated signal its amplitude follows the Rican distribution [18], and amplitude scintillation follows the Nagakami distribution. The difference operator of eqn (7) will eliminate the slow amplitude change caused by the background ionosphere, and the remaining is noise, which can be considered to be a Gaussian process with mean of m∆A0 and variance d A2 σ ∆2A , d A is a proportional factor related to the antenna gain decay for low satellite elevation angles. When scintillation occurs, ∆A is also modeled as a Gaussian process with mean m∂A1, which is proportional to the amplitude scintillation ∂A:  N(m , d 2σ 2 ) ∆A 0 ∆A ∆A ∝  2 2  N(m∆ A1 ,d σ ∆A ) no scint.  with scint. (8) 4.2 Wavelet Decomposition of Scintillation Signals We choose a low-order, orthonormal and compactlysupported wavelet basis (Daubechies wavelet of order 4) as the tool for scintillation signal decomposition, see Appendix A. The ionospheric scintillation signals can be decomposed as: j1 f (t) = ∑ c j 0 (k)ϕ j0, k (t) + ∑ ∑ d j (k)ψ j ,k (t) k k j=j0 (9) where j0 is usually chosen to give the wavelet description of the slowly changing or longer duration features of the signal, and the finest scale j1 is determined by the signal-sampling rate. If the sampling rate is high enough, and the scaling function behaves like delta function at fine scale j1, then cj1(k) can be considered as equals to the samples of f(t), and filter bank algorithms can be used to compute cj(k) and dj(k) at coarse scales: c j (k) = ∑ h(n − 2k)c j +1(n) (10) d j (k) = ∑ h1(n − 2k)c j +1(n) (11) n n Because eqn (10) and eqn (11) are linear operations, c j and d j are Gaussian processes: c j : N (mcj , σ cj2 ) (12) d j : N (m dj , σ dj2 ) (13) where m cj (k) = ∑ h(n − 2 k)m cj +1(n) (14) m dj (k) = ∑ h1(n − 2k)mcj +1(n) (15) n H 0 : X : N [m0 , σ 2 ] ~ no scintillation (19) H 1 : X : N [m1 , σ 2 ] ~ with scintillation (20) X is a Gaussian process, m0 and σ2 are known, but m1, which is related to the scintillation signal, is unknown, hence it is a two-sided detection with simple H0 and composite H1: H0 : θ = θ0 versus H1 : θ = θ1 ≠ θ0 (21) where θi = mi , i = 0, 1. Because there is no apriori probability of scintillation occurrence available, the Neyman-Pearson detectors are proper choice for the problem, see Appendix B. After some computation involving the logarithm of the likelihood ratio L(x), the test can be derived as: X > X t ~ choose H1 (22) X ≤ X t ~ choose H0 (23) The threshold value X t for the test is set by: n and: Pf = 2 ∫ σ cj2 (k) = σ cj2 +1 ∑ h2 (n) Xt /σ n =σ 2 cj +2 ∑ h (n) = L = σ 2 ∞ 2 (16) −( Xt −m1 ) / σ −∞ where σ2 is the noise variance of the phase or amplitude scintillation observables. Similarly: σ dj2 (k ) = σ 2 (17) because for orthonormal wavelet decomposition the following relation exists [5]: ∑ h (n) = ∑ h (n) = 1 2 n 2 1 (18) n 4.3 Ionospheric Scintillation Detectors Ionospheric scintillation detection is a binaryhypothesis-testing problem. Let X denote the phase or amplitude scintillation observables ∆Φ and ∆A, or their wavelet coefficient d j . According to eqn (6) and eqn (8), there are two physical system models which generate X: (24) and the detection probability is then computed as: Pd = ∫ n (2π )−1/ 2 exp(−u 2 / 2)du +∫ ∞ ( X t − m1 ) / σ (2π ) −1 / 2 exp(−u2 / 2)du −1/ 2 (2π ) (25) 2 exp(−u / 2)du The test is usually called the UMPU (Uniformly Most Powerful Unbiased) test [18], which has the largest detection power among all such two-sided and composite tests with specified P f . ‘Unbiased’ means that, for any value θ of the unknown parameter, the probability of false alarms is less than the detection probability. The detection probability is proportional to the signal-to-noise ratio m1 − m0 / σ , when m1 = m0 , Pd = Pf , which is the worst case. The scintillation detection can be carried out effectively using the wavelet-decomposed scintillation signal, and one UMPU detector is formed for each layer:  1 with scintillation   Hj =   0 no scintillation  (26) where H j is the j-th layer binary test of the wavelet coefficient d j , Finally, a multiple-binary test is formed, which is the union of each layer detector:  1 with scintillation   HM = U H j =   0 no scintillation  j =1 M (27) 4.4 Scintillation Tracking Sufficient statistics χ n2 /σ 2 of phase and amplitude scintillation observables accumulated in time, which follows the χ 2 distribution, can be used as the test statistics for scintillation time-lasting : n χn = ∑ X i 2 2 (28) i=1 and the hypothesis is:  H : χ 2 (µ 2 /σ 2 ,n) 2 2 0 χ n / σ =  0 2 2 2  H1 : χ (µ1 / σ , n) scint   with scint  no (29) 4.5 Scintillation Depth Indication Phase and amplitude scintillations are usually nonstationary, and their apriori probability distributions are unknown. In practice, especially for real-time applications, it is assumed that they are stationary and ergodic for short time intervals (typically a few minutes), and S 4 and σ δΦ can be approximated by time-averaged windowed scintillation measurements, rather than ensemble-averaging. As membership functions of the fuzzy sets are primarily subjective in nature, dependent on application-specific criteria [20], the linear function is chosen as the membership function of the linguistic variables because a linear relationship exists between σ δΦ and phase scintillation depth, and between S 4 and amplitude scintillation depth when S 4 < 0.7 and no Fresnel filtering and saturation effects occur. For ‘weak’ amplitude scintillation, the membership function is taken account of S 4 < 0.4 when the Rytov weak scintillation model is in effect. The membership functions of amplitude scintillation can then be written as: where µ n µ2i = ∑ < X j >2 , i=0,1 (30) j =1 and Xj A strong  S4 − 0.55 if  =  0.7 − 0.55 1 if   0.55 ≤ S4 < 0.7  S4 ≥ 0.7  (33) is the phase or amplitude scintillation observables ∆A or ∆Φ . µ χ 2 has a monotone-likelihood ratio. That is the larger the χ n2 /σ 2 is, the more probable the H 1 becomes. According to the Karlin-Rubin theorem [19] for the one-sided binary test µ12 = µ 02 versus µ12 > µ 02 , there is a UMP (Uniformly Most Powerful) detector: H 0 : χ n2 / σ 2 ≤ χ 02 (31) H 1 : χ n2 / σ 2 > χ 02 (32) where χ 02 is determined with respect to the specified false alarm probability. µ 0 are either zero or very near zero and change very slowly, hence it can be assumed to be stationary for short time periods, and can be estimated in real-time. The detection power of the UMP detector is proportional to the signal-to-noise ratio of µ12 − µ 02 / σ 2 . A weak  0.45 − S4 if  =  0.45 − 0.3  1 if   0.3 ≤ S4 < 0.45  (34) STA ≤ S4 < 0.3  where STA is a threshold value, which can be obtained from the threshold value of false alarm probability for the amplitude scintillation detector, or chosen subjectively. {( ) ( A A A µ medium = 1 − µ strong ∨ 1 − µ weak )} (35) where ∨ means fuzzy union. The indication of amplitude scintillation depth µ A can be derived from the membership functions of the arguments: A A A µ A = µ weak ∨ µ medium ∨ µ strong (36) Fig.1 membership fun. for amplitude scint. indication Fig.1 illustrates the membership functions. It can be A seen that when S 4 = 0.6999 , µ A = µ strong = 0.9993 , Fig.3 PRN31 elevation angle which means that the scintillation has 99.93% possibility of belonging to strong scintillation, or alternatively, the degree of ‘strong’ of the scintillation is 0.9993 or 99.93%. Similarly, the membership scintillation can be written as: µ P strong µ P weak functions of phase  σ δφ − 0.75  if =  1.0 − 0.75  1 if   0.75 ≤ σ δφ < 1  (37) σ δφ ≥ 1   0.45 − σ δφ  =  0.45 − 0.2  1   0.2 ≤ σ δφ < 0.45  σ TP ≤ σ δφ < 0.2  if if Fig.4 PRN31 L1 phase signal-to-noise ratio Fig.5 PRN31 L1 phase amplitude scint. observable (38) where σ TP is a threshold value, which can be obtained from the threshold value of false alarm probability for the phase scintillation detector, or chosen subjectively, and { P P P µ medium = (1 − µ strong ) ∨ (1 − µ weak ) } (39) Fig.6 PRN31 L1 phase minus L2 phase The dependency on satellite elevation angle of C / No , and the amplitude and phase scintillation observable, is apparent. For detection and tracking, we simply assumed that the phase scintillation observable coefficient is C=1. Fig.2 membership function for phase scint. indication 5. DATA PROCESSING Dual-frequency phase and its C / N 0 measurements were collected by a NovAtel-GPScard receiver, sited at − 3 0 58 ' 41".08 N, 119 0 38 ' 59." 59 E, during the Spring Equinox period of 1998, with a sampling interval of 0.5Hz. Fig.4 and Fig.6 illustrate typical examples of phase and amplitude scintillation. From the plots it can be seen that at epoch 4161 C/No has a 7 dB-Hz decay, due to the amplitude scintillation, and both L1 and L2 PLL lose lock. When the elevation angle is less than 10 deg., C/No is less than 38 dB-Hz, due to the low antenna gain, and PLL constantly lose lock. Fig.7 to Fig.12 illustrate the 5-layer wavelet decomposition of the phase scintillation observable. The double-straight lines are the threshold values of 0.0075 for the UMPU detector, which corresponds to σ δΦ =0.24 rad. According to eqn (34) and Fig.1, it has a degree of 0.84 for weak scintillation. The probability of false alarm is 0.0075, and the standard deviation of the observable is 0.0015 m, which is equivalent to 0.05 rad.. scintillation signal power is quite uniformly distributed at each layer, hence the signal exhibits a narrow-band white noise nature. Fig.7 PRN31 phase scint. observable Fig. 8 PRN 31 L1 phase scint. wavelet coef. of layer 1 Fig.9 PRN31 phase scint. Wavelet coef. of layer 2 Fig.10 PRN31 phase scint. Wavelet coef. of layer 3 Fig.11 PRN31 phase scint. Wavelet coef. of layer 4 Fig.12 PRN31 phase scint. Wavelet coef. of layer 5 From the layer plots and eqn (27) it can be seen that the phase scintillation began at epoch 3553. The phase Fig.13 to Fig.16 shows the wavelet decomposition of the amplitude scintillation observable. The threshold value of 0.5, which is equivalent to S 4 = 0.01 is also set for a probability of false alarm of 0.05. The standard deviation of the observable is 0.1dB-Hz. Fig. 13 PRN31 L1 C/No wavelet coef. of layer 1 Fig.14 PRN 31 L1 C/No wavelet coef. of layer 2 Fig. 15 PRN31 L1 C/No wavelet coef. of layer 3 Fig.16 PRN 31 L1 C/No wavelet coef. of layer 4 Fig.17 PRN 31 L1 C/No wavelet coef. of layer 5 From the Figures it can be seen that the amplitude scintillation power is mainly concentrated in the first three layers, approximately corresponding to a frequency range of 0.125 to 0.5Hz, and the first occurrence of scintillation is at about epoch 3035. Fig. 18 and Fig. 19 show the χ 2 statistics of the phase and amplitude scintillation for scintillation tracking. The degree of freedom is 30, corresponding to 1 minute data accumulation. The threshold values are also set by assuming a false alarm probability of 0.05, which are 0.002 and 0.45 for the phase and amplitude respectively. Fig.18 PRN31 phase scint. observable χ 2 statistics Fig. 21 PRN 31 S 4 value 6. CONCLUDING REMARKS Ionospheric scintillation features can be effectively and compactly exposed using the wavelet transform and an analysis in the time-frequency domain. The scintillation signal features can be detected reliably using multilayer binary statistical testing. Wavelet transform also offers the possibility for scintillation mitigation, and scintillation-free signals can be reconstructed by smoothing out the detected scintillation feature at different scales. The orthogonal wavelet transform creates a zero-mean process, and makes testing and algorithm development simpler. Linear fuzzy membership functions can offer a userfriendly and more reasonable expression of scintillation depth. The idea can also be expanded to detection tracking problems. The hypothesis H 0 and H 1 are no longer crisp sets, and have some overlap, hence fuzzyprobability or possibility-probability decision making is attractive for future applications. Fig.19 PRN31 amplitude scint. observable χ 2 statistics The approximate values of S 4 and σ δΦ are computed for each half-minute, considering the amplitude scintillation energy is concentrated in wavelet coefficients of first three layer decomposition. Fig.20 and Fig.21 illustrate the history of approximate values of S 4 and σ δΦ . The amplitude scintillation is ‘weak’, and the phase scintillation involves whole ‘weak’, ‘medium’ and ‘strong’ sets. Fig. 20 PRN31 phase scint. standard deviation REFERENCES [1] S. Basu & B. K. Khan, “ Model of equatorial Scintillation from in-situ Measurements”, Radio Sci., Vol. 11, 1976, pp.821-832. [2] D. J. Fang, “C-band Ionospheric Scintillation Measurements at Hong Kong Earth Station during the Peak of Solar Activities in Sunspot Cycle 21”, Proc. IES, Alexandria, VA, 1981. [3] J. Aaron, S. Basu, “Ionospheric Amplitude & Phase Fluctuations at the GPS Frequencies”, ION94, Sat. Div., Salt Lake City, Utah, 1994, pp.1569-1578. [4] C. Chui, “ Wavelets: A Mathematical Tool for Signal Analysis”, SIAM, Philad., 1997. [5] C. Burrus, et al, “Introduction to Wavelets & Wavelet Transform”, Prentice Hall, NJ, 1998. [6] S. Basu, M, C. Kelley, “ A Review of Recent Observations of Equatorial Scintillations & their Relationship to Current Theories of F-Region irregularities Generation”, Radio Sci., Vol. 14, 1979, pp.471-. [7] J. Aarons, “ Global Morphology of Ionospheric Scintillations”, Proc. IEEE, Vol.70, No. 4, April 1982, pp.360-378. [8] S. Basu, W. B. Hanson,” The Role of in-situ Measurements in Scintillation Modelling”, Proc. IES, Alexandria, VA, 1981. [9] K. Yeh, C. Liu,” Radio Wave Scintillations in the Ionosphere”, Proc. IEEE, Vol. 70, No. 4, April 1982, pp.324-359. [10] R. K. Crane, “Ionospheric Scintillation”, Proc. IEEE, Vol. 65, No. 2, Feb. 1977, pp.180-199. [11] M. Nakagami, “Statistical Methods”, Radio Wave Propagation, W.C. Hofmann, Ed., Elmsford, NY, Pergamon, 1960, pp.3-36. [12] E. Fremouw, et al., “ On the Statistics of Scintillating Signals”, Journal of Atmospheric and Terrestial Physics, Vol. 42, 1980, pp.717-731. [13] S. Pullen, et al, “Apreliminary Study of the Effect of Ionospheric Scintillation on WAAS User Availability in Equatorial Regions”, ION GPS-98, Sept. 15-18, 1998, Nashvile, Tennessee, pp.687-699. [14] S. Pullen, et al., “ A Preliminary Study of the Effect of Ionospheric Scintillation on WAAS User Availability in Equatorial Regions”, ION 98, Sept. 15-18, 1998, Nashville, Tennesse, pp.687-699. [14] J. Spilker, “ GPS Signal Structure & Performance Characteristics”, Global Positioning System, Inst of Navigation, Vol. 1, 1980, Washington, pp.29-54. [15] M. Knight, A. Finn, “Ionospheric Effects on GPS”, DSTO, Australia, 1997. [16] E. Fremouw, et al, “Complex Signal ScintillationsEarly Results from DNA-002 Coherent Beacon”, Radio Sci., Vol. 13, 1978, pp.167-187. [17] L. Wanninger, “Der Einfluss der Ionosphare auf die Positionierung mit GPS”, ISSN 0174-1454, Universitat Hannover, 1994. [18] R. McDonough, A. Whalen, “ Detection of Signals in Noise”, Academic Press, NY, 1995. [19] L. Scharf, “Statistical Signal Processing”, Addison-Wesley Pub. Co., Massachusetts, 1991. [20] H. Zimmermann, “Fuzzy Set Theory & its Application”, Kleuver-Nijhoff Pub., Boston, 1985. APPENDIX A: WAVELET DECOMPOSITION Supposing f(t) ∈ L2(R) (the space of square integrable function), L2 can be expressed as [intro]: L2 = Vj ⊕ Wj +1 ⊕ Wj +2 ⊕L where V j = Span(ϕ j , k ) (A1) ⊕ means direct sum, the overbar means the closure of the space expanded by the basis function. ϕ and ψ are scaling and wavelet functions at scale j: ϕ j ,k (t) = 2 ψ j , k (t ) j/ 2 ϕ(2 j / 2 t − k) (A2) = 2 j / 2ψ ( 2 j / 2 t − k ) (A3) ϕ (t) = ∑ h(n))2ϕ(2t − n) (A4) ψ (t) = ∑ h1 (n))2ϕ(2t − n) (A5) h1 (n) = (−1) n h( N − 1 − n) (A6) n n where N is the support of ϕ, and h(n) and h1(n) correspond to the impulse response of low-pass and high-pass FIR (Finite Impulse Response) filters for scaling and wavelet function respectively. APPENDIX B: NEYMAN-PEARSON DETECTOR: The Neyman-Pearson detection criteria is [] max P(D1 / H1 ) = max ∫R1 p1 (x)dx (B1) P(D1 / H 0 ) = ∫R1 p0 (x)dx ≤ Pf (B2) P( D1 / H 1 ) = 1 − P( D0 / H 1 ) = Pd (B3) where D0, D1 are decisions that H0 or H1 is in effect, Pf is the false alarm probability, Pd is the detection probability, R1 is the region of the X space in which D1 is in effect, and p0 and p1 are the densities of X conditioned on H0 and H1.. The Neyman-Pearson detectors are used so as to choose a false alarm (type 1 error) as large as we are willingly to tolerate and seek to minimize the probability of missed detection (type 2 error) under that limitation, or equivalently maximize detection probability while maintaining the false alarm probability at most at the specified level. The NeymanPearson detection lemma can be written as a likelihood ratio: k W j = Span(ψ k j ,k ) if L(x) = p1(x)/p0(x) > λ0 choose H1 (B4) if L(x) = p1(x)/p0(x) ≤ λ0 choose H0 (B5)