Coherent Radar Imaging

Transcription

1 Coherent Radar Imaging Ronald F. Woodman Jicamarca Radio Observatory Instituto Geofisíco del Perú Acknowledgments:Jorge L Chau, David Hysell, Erhan Kudeki

2 Scope Coherent Radar Imaging is the outgrowth of the more general Radar Imaging technique Radar Imaging includes: SAR Imaging Planetary imaging Georadar (underground) Meteorological imaging Coherent radar imaging Ionospheric irregularities Upper atmospheric turbulence Radar Imaging is, in turn, part of a broader technique: Radio Imaging which includes radio imaging in Astronomy. All these techniques are similar. We will limit ourselves to the topic of the title.

3 Scope (continued) Nevertheless, the most fundamental problems are common. Thus, coherent radar imaging borrows from the advances made in the other applications, especially from Astronomical Radio Imaging, which precedes radar imaging by a couple of decades. We hope some of the advances made by coherent radars will benefit the other techniques We will try to answer the questions: How is it done? What can be used for? What has produced? (Include a few examples)

4 Peculiarities of Coherent Radar Target is three (space) dimensional Target changes in time in two scales short defining color (frequency spectrum) long, the scale of non-stationary Target is a non-stationary nonhomogenous 4-dimensional stochastic process Relatively small number of independent samples available for averaging

5 Coherent radar imaging techniques Scanning Imaging with multiple-antenna arrays. Filled arrays Sparsely populated apertures Interferometers

6 Imaging by scanning Scanner analog Slit camera Images by scanning (Jicamarca) Sp-F irregularities E Region irregularities 150 km echoes Mesospheric echoes Stratospheric and Tropospheric Limitations

7 Slit Camera

8 Imaging by scanning Scanner analog Slit camera Images by scanning (Jicamarca) Sp-F irregularities E Region irregularities 150 km echoes Mesospheric echoes Stratospheric and Tropospheric Limitations

9 The Jicamarca Radio Observatory

10 ESF echoes (from Woodman and Chau [2001])

11 140 Equatorial Electrojet (a) SNR E (db) 18.0 Range (km) :45 18:50 18:55 19:00 Day: 19-Nov [Chau and Hysell, 2004]

12 150 km echoes

13 Mesosphere

14 Stratosphere and Troposphere

15 Slit-camera Analogy and Problems used with permission

16 Imaging from multi-antenna arrays (Diffraction theory) Filled aperture array Camera analog Truncation (aperture not large enough) Inversion of truncated visibility Sparsely populated aperture Non redundant spacing Sparsely sampled visibility Inversion Interferometry

17 F( q) B( q) F( q) F*( q) i F( ) = qix q f( x) e dx L >> D F( q ) f ( x) B( q) V( r) e q = {sin θ1,sin θ2} x = { kx, kx } 1 2 f ( x) ixiθ kernel θ e iriθ V( r) f( x) f *( x+ r) D

18 D f ( x) V () r θ F( q) f ( x) L >> D B( q) V( r) F( q) B( q)

19 F( q) f( x) and B( q) V( r) Where B( q) F( q) F*( q) and V( r) f( x) f *( x+ r)

20 Actually: B( θ ) Diagonal[ B( θθ, ) F( θ) F* ( θ )] and ixiθ B( θ,θ ) dxdx e V( x, x ) e = + ix iθ V( r) f( x) i f * ( x+ r) = DiagonalAverage[ V( x, x ) f( x) f where r = x x * ( x+ r) ] B V( xx, ) ( θ, θ )

21 Imaging from multi-antenna arrays (Diffraction theory) Filled aperture array Camera analog Truncation (aperture not large enough) Inversion of truncated visibility Sparsely populated aperture Non redundant spacing Sparsely sampled visibility Inversion Interferometry

22 Analogy with an pin-hole camera Object Plane B(θ ) True Brightness f( x) F( θ ) B( θ ) V( r) Aperture Plane Image Plane V () r Vˆ( r) True Visibility f Meassured Visibility ˆ( x) = a( x) f ( x) V ˆ( r ) = a( x ) a ( x+r ) V() r fˆ( x) Fˆ ( ˆ θ ) ˆ( 2 Bˆ( θ ) = A ( Vˆ θ( ) r ) B( θ) Estimated BA ˆ 2 (( θ ) θ ) Brightness a( x) a( x+ r) (from Woodman [1997])

23 Analogy with an optical camera i, j ix iθ i j M { M } = { e } F f fˆ Vˆ = a a MiB ˆ = F Mif ˆ Fˆ Bˆ = A 2 B (from Woodman [1997]) In radar imaging a can be an arbitrary complex vector

24 Butler Matrix Butler and Lowe, 1961 FFT Algorithm Cochran et al., 1967

25 Imaging Problem: Given: Vˆ = a a MiB or Bˆ = A 2 B Find a B estimate that best agrees with B under valid constrains. A typical inversion problem

26 Frequency spectra information (Color) f( x) is actually f( x, t) fˆ is Fourier transformed into fˆ x, t x, ω Cross-spectra, x, ω fˆ fˆ x, ω x, ω, is evaluated at (4) different ω-bands Each band is treated independently as explained Four B, corresponding to four ω bands, are evaluated. 3 colors (BGR) are assigned to 3 center ω-bands Color saturation is an indication of narrow spectral width

27 Imaging Problem: Given: Vˆ = a a MiB or Bˆ = A 2 B Find a B estimate that best agrees with B under valid constrains. A typical inversion problem

28 Inversion Techniques-A classification intent: Let La = scale size of a a and rc = the scale size of V, we can then consider the following cases: case action 1) Filled aperture L Use ˆ a > rc B as a good estimate of B Divide ˆ 1 V by a a ( 0) and then M (deconvolve). L Divide ˆ a rc V by a a ( 0), extrapolate 1 and then M (deconvolve). Extrapolate using MaxEnt 2) Sparse aperture Use Capon, Clean, deconvolution and Multiblob models Use MaxEnt th st nd 3)Interferometer La rc Evaluate 0, 1 and 2 moments.

29 Brightness and visibility used for the examples that follow V r F θ

30 Inversion Techniques-A classification intent: Let La = scale size of a a and rc = the scale size of V, we can then consider the following cases: 1) Filled aperture 2) Sparse apertur e L L a a 3)Interferometer L r Evaluate 0, a case action > r Use ˆ c B as a good estimate of B Divide ˆ 1 V by a a ( 0) and then M ( deconvolve). r Divide ˆ c V by a a ( 0), extrapolate 1 and then M ( deconvolve). Extrapolate using MaxEnt Use Capon, Clean, deconvolution and Multiblob models Use MaxEnt c 1 and th st nd 2 moments.

31 Brightness and visibility used in the following examples Fθ Vr

32 Case L a > r c ˆ B,B Vˆ = a a MiB ˆ V,V

33 Deconvolved image Fθ Vˆ MiB = a a for a a 0 ˆ -1 V B= M i a a Vr

34 Case L a < r c ˆ B,B Vˆ = a a not sufficient Divide by a a for x's such that a a 0 Extrapolate MiB V ˆ ( deconvolve) ˆ V,V

35 Case: sparse array ˆ B,B ˆ V,V Vˆ = a a MiB not sufficient Use band-limited and positive-definite properties of B

36 Case: sparse array B,Bˆ ˆ V,V ˆ Force Correct Inverse Iterate Transform After Divide Starting to some measured! transform zero by point back a*a iterations out-of values to band a new and B negative B s

37 Inversion Techniques-A classification intent: Let La = scale size of a a and rc = the scale size of V, we can then consider the following cases: case action 1) Filled aperture L Use ˆ a > rc B as a good estimate of B Divide ˆ 1 V by a a ( 0) and then M (deconvolve). L Divide ˆ a rc V by a a ( 0), extrapolate 1 and then M (deconvolve). Extrapolate using MaxEnt 2) Sparse aperture Use Capon, Clean, deconvolution,multiblob models Use MaxEnt th st nd 3)Interferometer La rc Evaluate 0, 1 and 2 m oment s.

38 Maximize Entropy, MaxEnt With the following constrains: S = B ln( B / B ), θ θ θ θ The normalized measured visibility, xx, xx, for each antenna pair, conforms with the FT of the normalized brightness distribution, B / B, plus an estimation error, e θ θ xx,.. θ Errors ( e x, x ) are allowed, but bounded to their estimated theoretical value, in the maximization process. Correlation between errors (After Hysell&Chau, submitted 2005) is taken into account. θ V ˆ / P,

39 Inversion Techniques-A classification intent: Let La = scale size of a a and rc = the scale size of V, we can then consider the following cases: case action 1) Filled aperture L Use ˆ a > rc B as a good estimate of B Divide ˆ 1 V by a a ( 0) and then M (deconvolve). L Divide ˆ a rc V by a a ( 0), extrapolate 1 and then M (deconvolve) or use Capon. Extrapolate using MaxEnt 2) Sparse aperture Use Capon, Clean, deconvolucion, Multiblob models Use MaxEnt 3)Interferometer L a r c th st nd Evaluate 0, 1 and 2 moments.

40 Capon s method (from Capon [1969] Given V { fˆ fˆ } Capon's B { B } is given by: B c = i i where w { w } is such that B θ c θ x w V w θ, x x is a minimum, under the constrain ew i = 1 for every θ. Here, e { e i θ ix }are the sampled values, at x, of a unitary plane wave coming from θ. The constrains are satisfied by: B c 1 = eiv -1. ie { w } x, θ DFT Advantage: Solution involves a single Matrix inversion!

41 MaxEnt Examples Jicamarca Spread F Electrojet Aurora QPE Trospheric Turbulence

42 SpF, Jicamarca Observatory ±50 m/s [Hysell et al., 2004]

43 SpF, Jicamarca Observatory ±150 m/s [Hysell et al., 2004]

44 Imaging at Jicamarca: 2D Imaging EEJ at Twilight [from Chau and Hysell,2004] ±50 m/s

45 Daytime Electrojet over Jicamarca [Hysell, Chau et al., PC]

46 QP Echoes over Puerto Rico Arecibo Puerto Rico ±300 m/s [Hysell et al., 2004]

47 Aurora, Alaska Hysell, p.c., 2005 [Bahcivan et al., 2005]

48 Aurora, Alaska Hysell, p.c., 2005 [Bahcivan et al., 2005]

49 Imaging at Jicamarca (9): 2D Imaging 150-km echoes [Chau, Kudeki, Hysell, PC, 2005]

50 Imaging at Low Latitudes (1): Piura (14± Dip) [Chau, PC, 2005]

51 QP echoes over Piura (14± Dip) [Chau, PC, 2005]

52 Capon Method Tropospheric Imaging at MU (from Palmer et al. [1998]) Irregularities are almost isotropic. Use of Capon method Images of the aspect sensitivity (5 min integration). Brightness intensity is color-coded. Tropospheric Images Fourier Capon Improved resolution! Images of rain obtained through Doppler sorting

53 Tropospheric imaging with the Provence ST VHF radar (from Hélal et al. [2000]) Use of very wide tx/rx beam widths ( 60 o ) and 8/16 rx channels. Although, only one physical rx is used. Rx channels are multiplexed with a highcommunication rate switch. Observations of very wide horizontal stratified structures using Sequential PBS, Capon and MUSIC. Fourier images [SNR( ) and Doppler ( )]

54 3D Imaging at Jicamarca (from Chau and Woodman [2000] and poster) Note a meteor echo in the 3D image. The tropospheric images do not show continuity with height. Not much gain in information is obtained by using more than 3 antennas, at least at the time of the experiment and/or with the narrow field of view employed. Receiving Configuration Tropospheric Images 3D Equatorial Electrojet image

55 Inversion Techniques-A classification intent: Let L = scale size of a a and r = the scale size of V, we can then consider a the following cases: c case action 1) Filled aperture L Use ˆ a > rc B as a good estimate of B Divide ˆ 1 V by a a ( 0) and then M (deconvolve). L Divide ˆ a rc V by a a ( 0), extrapolate 1 and then M (deconvolve) or use Capon. Extrapolate using MaxEnt 2) Sparse aperture Use Capon, Clean, Multiblob models Use MaxEnt 3) Inter Evaluate 0, th st nd ferom eter La rc 1 and 2 mome nts.

56 x θ x Interferometry xx, and it can be shown that: P = V(0) where P Power B( θ) dθ θ = φ( r ) / r θ Mean angle of arrival θb( θ) dθ 2(1 V( r ) / P σ = σ Width ( θ θ ) B( θ) dθ 2 d 2 2 rd These properties can be generalized to 2-Dimensions. *( d d Woodman and r d Cross-correlator V Guillen,1974) In any FT pair, like B( θ ) V( r), the derivatives, evaluated at the origen of one are proportional to the moments of the same order of the other. If the distance r = x x r, d where r is the caracteristic size of V c V r e iφ ( r) xx, = ( ). Then, V can be expanded in a Taylor series, xx, Coherence c

57 Interferometer Applications Meteor head echoes Aspect sensitivity Magnetic field inclination First electrojet images Electrojet drifts

58 Interferometer : Meteor heads radians&velocities [Chau and Woodman, 2004]

59 Interferometry at Jicamarca (6) Aspect Sensitivity Configuration Receive signal on Various 64 th s Transmit on East and West Quarters Receive common Signal in 1/64 th [e.g. F. Lu, 2005; Ph.D. Thesis]

60 Interferometry at Jicamarca (7) EEJ Aspect Sensitivity Measurements [Kudeki and Farley, 1989] [e.g. F. Lu, 2005; Ph.D. Thesis]

61 Woodman, 1971

62 Interferometry at Jicamarca (2) Measurements of Magnetic Field Inclination Using Incoherent scatter theory, combined with N-S Interferometer, Woodman [1971] was able to tome measure the inclination of the magnetic field above Jicamarca with 1 min of arc accuracy. At the time, models were off by ~1 o. [e.g. Woodman, 1971]

63 Thank you