The Roles of Diffractive and Refractive Scattering in the Generation of Ionospheric Scintillation Charles S. Carrano

Transcription

1 The Roles of Diffractive and Refractive Scattering in the Generation of Ionospheric Scintillation Charles S. Carrano Boston College, Institute for Scientific Research ISR Seminar Series, Boston College 2 December 2014

2 Purpose of this talk: characterize the roles of refractive and diffractive processes that contribute to the scintillation of transionospheric signals, so that their signatures can be identified in experimental measurements and used to infer the statistical structure of the random medium. Outline: Propagation past sinusoidal and general phase screens (tutorial) Propagation past power law phase screens (tutorial / new) The statistical theory of scintillation (new) Data analysis and Irregularity Parameter Estimation (new) 2

3 Scintillation Physics: A Simple Picture (Groves) τ d = R/ c+ rc e N 2π f tot 2 FROM SATELLITE N δφ = 2 π fr / c re c f tot ΔNe IONOSPHERE Radio Wave Interference Pattern V DRIFT Ntot = Ne( z) dz EARTH S SURFACE Phase variations on wavefront from satellite cause diffraction pattern on ground Interference pattern changes in time and space User observes rapid fluctuations of signal amplitude and phase 3

4 Some Relevant Terminology Refraction is a change in direction of light due to a change in the physical properties (refractive index) of the medium through which it travels. To conserve energy and momentum the wave s phase velocity changes but its frequency remains constant. Diffraction is due to the inherent wave nature of light. It also takes the form of bending of the rays, but it is a manifestation of their constructive and destructive interference. It does not follow such simple laws nor take place on as large a scale as refraction. Electromagnetic Scattering is the process whereby light radiation is forced to deviate from a straight trajectory by one or more paths due to localized non-uniformities in the medium through which they pass. Fresnel scale - the transverse distance from the direct path for which the Huyghens wavelet is in quadrature with the resultant signal. Thus, a small phase change produced by irregularities this distant from the direct path produces a change in signal amplitude. Diffractive scattering is caused by irregularities smaller than the Fresnel scale. Refractive scattering is caused by irregularities larger than the Fresnel scale. References: Buttler, H. E., 1951; Budden, 1965; Booker, H. G. and MajidiAhi,

5 Focusing and Defocusing Figure from Dennis Knepp, 2008 URSI Mtg. 5

6 The Phase Screen Concept TX 3D random medium Equivalent screen RX An equivalent thin screen placed at the middle of an extended random medium can be constructed to exactly reproduce the amplitude and phase fluctuations on the ground. Essentially, the physical process of weak scattering from many of irregularities is replaced by a single (possibly strong) scattering at the phase screen. 6

7 Refraction and Diffraction (Simple Geometric Arguments) 7

8 Weak Scatter from a Sinusoidal Phase Screen Consider a thin screen that imparts a sinusoidal phase change: φ ( x ) = φ ( Kx ) 0 cos 1 to an incident monochromatic plane wave of unit intensity: ( ) ( ω ) E xzt,, = e i kz t The field just below the screen is: i 0 (,0) cos( Kx ) E + x = e φ ( ) e i kz ω t (to simplify notation, we ll drop the fast variation ) 0 2 π / K 1+ iφ cos( Kx) Briggs, 1975 and Hewish, 1989 z If the perturbation is small, ( φ0 1) x e we can Taylor expand exponential to first order: = 1+ x + This gives the weak scatter approximation for the field just past the screen: ( ) + φ0 + E x,0 1 i cos( Kx) incident scattered 8

9 Weak Scatter from a Sinusoidal Phase Screen Compare the scattered field E ( x,0) = iφ cos( Kx) s 0 with the sum of two plane waves propagating at angles ±θ to the zenith: s (, ) A i( kz cosθ+ kx sin θ) i( kz cosθ kx sin θ) E xz = e + e A 1 At the screen: these are equal if ( ) E x,0 = 2Acos( kxsin θ ) s A θ = K k = 1 iφ0 and sin / 2 θ E 0 2 π / K 1 2 iφ iφ 0 z We have decomposed the scattered field into its angular spectrum of plane waves The scattered plane waves must travel a longer slant distance than the incident wave. Hence phase differences accumulate as the waves propagate, and interference occurs. 9

10 Weak Scatter from a Sinusoidal Phase Screen Given the angular spectrum of plane waves at the screen, we can determine the field at any distance past the screen as the superposition of these plane waves: ( ) E x, z = i cos( kx sin θ ) e s = φ φ i φ i cos( Kx) e cos( Kx) e ikz cosθ ikz 1 K / k ikz e i K 2 2 k 2 2 z assuming K / k 1 k 2 2 K θ K k = φ 0 z + i z 2k 2k 2 2 cos( Kx) sin( K ) cos( K ) e ikz = + χ + χ + is If we express the total field past the screen as 1 E e is then we can identify the log amplitude and phase departure of the scattered field: χ 2 ( xz, ) = φ cos( Kx) sin( K z) Sxz 0 2k 2 (, ) = φ cos( Kx) cos( K z) 0 2k e e ikz ikz 10

11 Weak Scatter from an Arbitrary Phase Screen For weak scatter the problem is linear and we can express the scattered field beneath an arbitrary phase screen as a superposition of scattered sine (and cosine) waves: s ( ), = φ0( K) e i K 2 2 k E xz i e z ikx dk χ = 2 (, ) ( )sin( K ) 0 2k 2 (, ) ( )cos( K ) 0 2k ikx x z φ K z e dk = ikx S x z φ K z e dk Under these conditions the Spectral Density Functions (SDF) of χ and S are directly related (i.e. on a per wavenumber basis) to the SDF of the phase due to the screen: Φ = Φ χ S 2 2 ( K) sin ( K z) ( K) 2 2k Φ = Φ 2 ( K) cos ( K z) ( K) 2k φ φ Φ ( K) +Φ ( K) =Φ ( K) χ S φ As a consequence, the statistical properties of the medium are directly related to the statistics of measured amplitude and phase fluctuations on the ground. 11

12 Weak Scatter from an Arbitrary Phase Screen For weak scatter, the Spectral Density Function of χ and S are Fresnel filtered versions of the SDF of the variations in the screen: Φ = Φ χ 2 ( ) sin ( 2 K K z ) ( K ) 2k I χ ( K) φ I χ ( K ) z=350km Φ = Φ φ S 2 2 ( K) cos ( K z) ( K) 2k I S ( K ) I ( K ) S Kz/ k The amplitude fluctuations are high-pass filtered with cutoff L= 2λz (Fresnel scale) so they cannot be used to probe the large scale structure of the medium. The phase filter passes the large scale components. The amplitude and phase are notch-filtered as spectral components of the wave are alternately converted from phase to amplitude fluctuations with distance from the screen. 12

13 Fresnel Zone A Geometric Picture Plane wave TX The multipath signal (red) travels the extra path length D: 2 2 z ( ρ ) 2 D = ρ + z = z z + 1 z z RX ρ scatterer If ρ/z<<1 can expand in Taylor series: ( ) ρ D z + ρ z z = 2 2 z The first Fresnel zone defines the volume where wavefronts interfere constructively (D<λ/2). Fresnel zone radius: ρ = λz The Fresnel zone defines the most effective turbulent cell size in producing scintillation at distance z past the screen. Smaller cells contribute less to scintillation because of the weaker refractive index fluctuations associated with them, and larger cells do not diffract light through a large enough angle to reach the receiver at z. Driggers, 2003 This description applies in weak scatter only where the scattering angle θ K/k is independent of screen strength. In strong scatter, larger scales also contribute. 13

14 Strong Scatter from a Sinusoidal Phase Screen If the phase perturbation is larger (~1), we must keep higher order terms when expanding the exponential (let s start with 2 nd order): x 1 2 e = 1+ x+ x 2 E 0 With this approximation the field just after the screen is: 2 π / K ( ) φ E x,0 1+ iφ cos( Kx) cos ( Kx) θ 2θ z incident scattered E 0 Using a trigonometric identity (double angle formula) this can be written ( ) φ 4 0 φ E x,0 1 + i cos( Kx) cos(2 Kx) incident scattered Comparing the scattered field with the sum of 2 pairs of plane waves propagating at angles ± θ1, ± θ2 to the zenith gives the amplitudes and scattering angles: φ0 θ φ 8 0 θ2 A = i, sin = K / k A =, sin = 2 K / k 14

15 Strong Scatter from a Sinusoidal Phase Screen Given the angular spectrum of plane waves at the screen the field at any distance past the screen can be expressed as the superposition of these waves: s ikz cosθ1 1 2 ikz cosθ2 (, ) = φ cos( sin θ ) φ cos( sin θ ) E x z i kx e kx e ikz 1 K / k 1 2 ikz 1 (2 K ) / k 0cos( ) φ 4 0 cos(2 ) = φ i Kx e Kx e φ K 2 (2 K ) 2 2 k 2 2 k ikz i z 1 i z 0cos( ) e φ 4 0 cos(2 ) e i Kx e Kx The field at the ground now contains higher frequency content than the phase screen. This is a consequence of the non-linearity associated with strong scatter. 15

16 Strong Scatter from a Sinusoidal Phase Screen Let s revisit the wavefield just past the screen, with no limitation for screen amplitude φ 0 i 0 (,0) cos( Kx ) E + x = e φ E 0 Generating function for the Bessel function: i Kx n inkx φ cos( ) 0 e = i Jn( φ0) e n= J 2 π / K n ( φ0 ) z Comparing with sum of plane wave pairs at the screen ( ) E x,0 = 2 A cos( kxsin θ ) s n n n= 1 gives the amplitudes and scattering angles at the screen. Final result is a Fourier series: J n ( φ0 ) 2 2 iπ /4 in ( K /(2 k)) z inkx Exz (, ) = e Jn( φ0) e e n= sin θ n = ± ( nk / k) E 0 Spectral amplitudes Even though the screen has content at a single frequency, the intensity field is broadband 16

17 How Broadband is the Diffraction Pattern? J n (ϕ 0 =80) Order n The Fourier series effectively terminates at n ~ ϕ 0 The intensity varies on wavenumbers up to nk = ϕ 0 K (this is called the diffraction scale). The deeper the screen (i.e. the larger ϕ 0 ) the more broadband the diffracted field will be. 17

18 Compare with Refraction from a Sinusoidal Screen An observer close to the screen will see a single image refracted by an angle dictated by the local phase gradient θ r = 1 dϕ k dx = K k φ sin( Kx) 0 Far behind the screen at a distance z >> (Kθ r ), where the refracted rays reach the observer from many points in the screen, the observer sees multiple images of the source contained with an angular range set by the extreme value θ r K φ0 k This is consistent with the maximum scattering angle predicted by diffraction. However, the predictions of refraction are incorrect when the phase screen is weak. (Recall that in weak scatter θ K/k was independent of screen strength). 18

19 Strong Scatter from a Sinusoidal Phase Screen Phase screen Focal distance (k/φ 0 K 2 ) Reception plane Diffraction Refraction Caustic Refraction captures signal envelope but diffraction creates the small scale structure. 19

20 Still Stronger Scatter from a Sinusoidal Phase Screen Phase screen Focal distance (k/φ 0 K 2 ) Reception plane Diffraction Refraction Caustic 20

21 Power Law Phase Screens 21

22 In-Situ (1D) Power Law Spectra from C/NOFS PLP Rino, C. L., C. S. Carrano, and P. Roddy (2014), Wavelet-based analysis and power law classification of C/NOFS high-resolution electron density data, Radio Sci., 49, , doi: /2013rs The phase spectral index p is one larger than the in-situ spectral index, p (1) Observations generally fall in the range 2.5 < p < 4 22

23 Power Law Phase Screens Fully developed turbulence Shallow Screen p<3 Decaying turbulence Steep Screen p>3 These screen realizations have the same large scale structure, but the shallower spectra realization has more small scale structure. 23

24 Steep Spectra - Refraction Dominated Scintillation Phase screen Reception plane For this φ 0 and distance past the screen, refraction alone can explain the observed scintillation Diffraction Refraction At greater distances past the focal plane refractive focusing contributes less, whereas the small scale diffraction is always present. 24

25 Steep Spectra - Refraction Dominated Scintillation Lowpass filtered at Fresnel Frequency/4 Phase screen Reception plane Removal of small scale structure leaves pure refractive scattering. Diffraction Refraction The true S 4 index (red) has decreased slightly, which seems reasonable given that we have weakened the total perturbation. 25

26 Shallow Spectra - Diffraction Dominated Scintillation Phase screen Reception plane There is plenty of refraction, but the net effect is minimal (no focusing) Diffraction Refraction In this example, the reception plane is in the far field where only small-scale diffraction contributes Presence of multiple scale sizes each with different focal lengths causes focusing effects to average out. Many rays pass through each point on the ground, and we must account for their interference 26

27 Shallow Spectra - Diffraction Dominated Scintillation Lowpass filtered at Fresnel Frequency/4 Phase screen Reception plane As before, removal of small scale structure leaves pure refractive scattering. Diffraction Refraction The true S 4 index (red) has increased, since removal of small scale structure mitigates the diffraction and promotes focusing. 27

28 Statistical Theory of Scintillation 28

29 Phase Screen Theory Spectrum of intensity fluctuations: { } 2 ( ) ( ρf) Φ q = exp g r, q exp( iqr) dr I Fresnel scale: ρ F =(z/k) 1/2 Structure interaction term: ( ) = ΦI 1 2 g( r, r ) 8 q sin ( rq/ 2)sin ( r q/ 2)dq/ 2π Intensity correlation: 1 2π ( ) R( r) = Φ q cos( qr) dq I Scintillation index: S ( ) = 2π ΦI q dq 1 29

30 Two-Component Structure Model (Rino) Phase spectrum: < Φ δφ ( ) = p1 q, q q q Cp q p2 p1 q p1, q > q p 1 =0 gives an outer scale p 1 =p 2 gives an unmodified power law p 2 >p 1 gives a two-component spectrum p 1 Phase PSD p 2 q 0 q 30

31 Compact Strong Scatter Model (Rino) We use the Fresnel scale to normalize the phase and intensity spectra: P( µ ) =Φ ( ; ) δφ µρ / ρ = ( ) { } I( µ ) =Φ µ ; ρ / ρ = exp γ ( η, µ ) exp( iηµ ) dη I F F p1 U1µ, µ µ 0 F F p2 p1 p2 U2µ 0 µ, µ > µ 0 Normalized spatial wavenumber: µ = qρ F Normalized turbulent strength parameters: p1 1 p2 p1 p2 1 1 = pρf, 2 = p 0 ρf U C U C q Next, define U* as the normalized phase spectral power at the Fresnel scale U * U = U, q ρ 1 1 0, q ρ < F F Then parameters p 1, p 2, µ 0, and U * specify all solutions for 2-component spectra (i.e. different combinations of perturbation strength, propagation distance, and frequency produce exactly the same results). 31

32 Single-Component Power-Law with p=2.5 Intensity PSD Increasing U (screen strength or distance) Non-dimensional Wavenumber Significant departures from power law behavior occur when the scatter is strong. In particular, note the low frequency enhancement and spectral broadening. 32

33 Single Component Power-Law with p=3.5 Intensity PSD Increasing U (screen strength or distance) Non-dimensional Wavenumber 33

34 Two Component Power-Law with p 2 =2.5 If we erode large scale irregularity structure, the low frequency enhancement is suppressed. The low frequency enhancement is a consequence of large scale refractive focusing. It is primarily responsible for driving S 4 above unity this is referred to as strong focusing. 34

35 Two Component Power-Law with p 2 =3.5 If we erode the large scale irregularity structure in this case, two things happen. As with the shallow slope case, the low frequency enhancement is suppressed. In the steeply sloped case, however, the spectral broadening at high frequencies is also suppressed. We can infer that focusing by large scale irregularity structure is responsible for the high frequency spectral broadening. This refractive scattering effect is commonly observed in late-night scintillation following the decay of small scale irregularities. 35

36 Development of the Scintillation Index (S 4 ) Unmodified power law Unmodified power law spectra with p>3 admit sustained quasi-saturation states with S 4 >1 Outer scale µ 0 =0.1 An outer scale suppresses strong focusing, driving S 4 to unity (from above if p>2) 36

37 Development of the Correlation Length Unmodified power law Outer scale µ 0 =0.1 For shallow spectra, correlation length is completely unaffected by presence of outer scale. For steep spectra, suppression of refractive focusing greatly increases correlation length. 37

38 Diffractive and Refractive Scattering at Ascension Island 38

39 Campaign at Ascension Island in March 2000 Geostationary satellites broadcasting radio signals at UHF (250 MHz) and L-band (1535 MHz) were monitored along nearly co-linear links. The UHF data were acquired using spaced antennas to measure the zonal irregularity drift. FLEETSAT F8 (250 MHz) INMARSAT 3-F2 (1535 MHz) Receiver: S, W, DEC km Side View RX 39

40 Intensity and Decorrelation Time March 2000 I I 40

41 Irregularity Parameter Estimation (IPE) Early evening bubbles (both UHF and L-band in saturation with S 4 ~1) UHF L-band When scatter is strong the intensity spectra deviate from power-law form (hence the irregularity parameters cannot be measured directly). Nevertheless, the irregularity parameters may still be inferred by non-linear least-squares fitting to the 4 th moment in the frequency domain. 41

42 Irregularity Parameter Estimation (IPE) Two hours later (UHF in strong focusing, L-band in weak scatter) UHF L-band Flat segment here suggests Φ ( q) q δφ 4 Because L-band is in weak scatter, in this case, we see direct evidence of the underlying two-component irregularity spectrum. Weak scatter theory predicts: 4 q Φδφ ( q), q q Iq ( ) Φ δφ ( q), q q f f 42

43 Reconciliation of S 4 at UHF & L-band Strong focusing, S 4 >1 Strong focusing, S 4 >1 43

44 Structure Model Parameters Focusing at UHF occurs more frequently as the break scale increases. 44

45 Universal Scaling Parameter, U * 45

46 Evolution of Irregularity Structure UHF L-band As local time increases, the Fresnel scale at UHF lies more fully in the steeply sloped region. This promotes strong focusing. 46

47 Conclusions (1) In weak scatter there is a direct relationship between scales in the diffraction pattern and the corresponding scales of phase variations in the random ionospheric medium. Since amplitude fluctuations are high-pass filtered they cannot be used to probe large the scale structure. Phase fluctuations are sensitive to large scale structure but can be difficult to interpret in strong scatter. When scatter is strong, large scale structures can contribute to the scintillation via refractive scattering. Efficiency of refractive scatter depends on the spectral index (p>3 is required for significant focusing) and proximity to the screen. For a given perturbation strength, focusing plays a decreasing role with increasing distance past the focal plane (which varies with scale) but the small scale diffraction is always present. 47

48 Conclusions (2) Early at night we often observe fully developed turbulence (p<3) in saturation (S 4 ~1). The presence of small scale structure suppresses the efficiency of refractive focusing. Late at night small scale features decay via diffusive processes, resulting in more steeply sloped spectra (p>3). This condition makes refractive focusing more efficient and produces S 4 values exceeding 1. At Ascension Island we observed 2-component spectra with largely unchanging slopes; the dominant effect was migration of the spectral break to larger scales over time. This caused the UHF signal to progressively feel more of the steeply sloped region (which admits strong focusing), and less of the shallow sloped region (which does not). 48