Abstract

A method for producing a radar-reflectivity map of the polar regions of the Moon or a planet from polar orbit with only the frequency shift of the reflected signals is described and simulated. A Radon transform of the reflectivity is obtained during multiple passes over the pole. Inversion of this Radon transform enables a map of radar reflectivity to be synthesized.

© 1997 Optical Society of America

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References

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  1. N. J. S. Stacy, “High resolution synthetic aperture radar observations of the Moon,” Ph.D dissertation (Cornell University, Ithaca, N.Y., 1993).
  2. J. K. Harmon, M. A. Slade, “Radar mapping of Mercury: full-disk images and polar anomalies,” Science 258, 640–643 (1992).
    [CrossRef] [PubMed]
  3. R. F. Jurgens, M. A. Slade, R. S. Saunders, “Evidence for highly reflecting materials on the surface and subsurface of Venus,” Science 240, 1021–1023 (1988).
    [CrossRef] [PubMed]
  4. D. O. Muhleman, B. J. Butler, A. W. Grossman, M. A. Slade, “Radar images of Mars,” Science 253, 1508–1513 (1991).
    [CrossRef] [PubMed]
  5. R. Gens, J. L. Vangenderen, “SAR interferometry—issues, techniques, applications,” Int. J. Remote Sensing 17, 1803–1835 (1996).
    [CrossRef]
  6. W. T. K. Johnson, “Magellan imaging radar mission to Venus,” Proc. IEEE 79, 777–790 (1991).
    [CrossRef]
  7. J. H. Thomson, J. E. B. Ponsonby, “Two-dimensional aperture synthesis in lunar radar astronomy,” Proc. R. Soc. London Ser. A 303, 477–491 (1968).
    [CrossRef]
  8. J. H. Taylor, “Two-dimensional brightness distributions of radio sources from Lunar occultation observations,” Astrophysics 150, 421–426 (1967).
    [CrossRef]
  9. D. O. Muhleman, “Radar scattering from Venus and the Moon,” Astron. J. 69, 34–41 (1964).
    [CrossRef]
  10. T. S. Durrani, D. Bisset, “The Radon transform and its properties,” Geophysics 49, 1180–1187 (1984).
    [CrossRef]
  11. G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).
  12. Y. Nievergelt, “Elementary inversion of Radon’s transform,” SIAM Rev. 28, 79–84 (1986).
    [CrossRef]
  13. J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integrralwerte langs gewisser Mannigfaltigkeiten,” Ber. Verhb. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262–277 (1917).
  14. L. A. Kunyansky, “Generalized and attenuated Radon transforms: restorative approach to the numerical inversion,” Inverse Probl. 8, 809–819 (1992).
    [CrossRef]
  15. S. J. Ostro, “Planetary radar astronomy,” Rev. Mod. Phys. 65, 1235–1279 (1993).
    [CrossRef]

1996 (1)

R. Gens, J. L. Vangenderen, “SAR interferometry—issues, techniques, applications,” Int. J. Remote Sensing 17, 1803–1835 (1996).
[CrossRef]

1993 (1)

S. J. Ostro, “Planetary radar astronomy,” Rev. Mod. Phys. 65, 1235–1279 (1993).
[CrossRef]

1992 (2)

L. A. Kunyansky, “Generalized and attenuated Radon transforms: restorative approach to the numerical inversion,” Inverse Probl. 8, 809–819 (1992).
[CrossRef]

J. K. Harmon, M. A. Slade, “Radar mapping of Mercury: full-disk images and polar anomalies,” Science 258, 640–643 (1992).
[CrossRef] [PubMed]

1991 (2)

D. O. Muhleman, B. J. Butler, A. W. Grossman, M. A. Slade, “Radar images of Mars,” Science 253, 1508–1513 (1991).
[CrossRef] [PubMed]

W. T. K. Johnson, “Magellan imaging radar mission to Venus,” Proc. IEEE 79, 777–790 (1991).
[CrossRef]

1988 (1)

R. F. Jurgens, M. A. Slade, R. S. Saunders, “Evidence for highly reflecting materials on the surface and subsurface of Venus,” Science 240, 1021–1023 (1988).
[CrossRef] [PubMed]

1986 (1)

Y. Nievergelt, “Elementary inversion of Radon’s transform,” SIAM Rev. 28, 79–84 (1986).
[CrossRef]

1984 (1)

T. S. Durrani, D. Bisset, “The Radon transform and its properties,” Geophysics 49, 1180–1187 (1984).
[CrossRef]

1968 (1)

J. H. Thomson, J. E. B. Ponsonby, “Two-dimensional aperture synthesis in lunar radar astronomy,” Proc. R. Soc. London Ser. A 303, 477–491 (1968).
[CrossRef]

1967 (1)

J. H. Taylor, “Two-dimensional brightness distributions of radio sources from Lunar occultation observations,” Astrophysics 150, 421–426 (1967).
[CrossRef]

1964 (1)

D. O. Muhleman, “Radar scattering from Venus and the Moon,” Astron. J. 69, 34–41 (1964).
[CrossRef]

1917 (1)

J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integrralwerte langs gewisser Mannigfaltigkeiten,” Ber. Verhb. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262–277 (1917).

Bisset, D.

T. S. Durrani, D. Bisset, “The Radon transform and its properties,” Geophysics 49, 1180–1187 (1984).
[CrossRef]

Butler, B. J.

D. O. Muhleman, B. J. Butler, A. W. Grossman, M. A. Slade, “Radar images of Mars,” Science 253, 1508–1513 (1991).
[CrossRef] [PubMed]

Durrani, T. S.

T. S. Durrani, D. Bisset, “The Radon transform and its properties,” Geophysics 49, 1180–1187 (1984).
[CrossRef]

Gens, R.

R. Gens, J. L. Vangenderen, “SAR interferometry—issues, techniques, applications,” Int. J. Remote Sensing 17, 1803–1835 (1996).
[CrossRef]

Grossman, A. W.

D. O. Muhleman, B. J. Butler, A. W. Grossman, M. A. Slade, “Radar images of Mars,” Science 253, 1508–1513 (1991).
[CrossRef] [PubMed]

Harmon, J. K.

J. K. Harmon, M. A. Slade, “Radar mapping of Mercury: full-disk images and polar anomalies,” Science 258, 640–643 (1992).
[CrossRef] [PubMed]

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

Johnson, W. T. K.

W. T. K. Johnson, “Magellan imaging radar mission to Venus,” Proc. IEEE 79, 777–790 (1991).
[CrossRef]

Jurgens, R. F.

R. F. Jurgens, M. A. Slade, R. S. Saunders, “Evidence for highly reflecting materials on the surface and subsurface of Venus,” Science 240, 1021–1023 (1988).
[CrossRef] [PubMed]

Kunyansky, L. A.

L. A. Kunyansky, “Generalized and attenuated Radon transforms: restorative approach to the numerical inversion,” Inverse Probl. 8, 809–819 (1992).
[CrossRef]

Muhleman, D. O.

D. O. Muhleman, B. J. Butler, A. W. Grossman, M. A. Slade, “Radar images of Mars,” Science 253, 1508–1513 (1991).
[CrossRef] [PubMed]

D. O. Muhleman, “Radar scattering from Venus and the Moon,” Astron. J. 69, 34–41 (1964).
[CrossRef]

Nievergelt, Y.

Y. Nievergelt, “Elementary inversion of Radon’s transform,” SIAM Rev. 28, 79–84 (1986).
[CrossRef]

Ostro, S. J.

S. J. Ostro, “Planetary radar astronomy,” Rev. Mod. Phys. 65, 1235–1279 (1993).
[CrossRef]

Ponsonby, J. E. B.

J. H. Thomson, J. E. B. Ponsonby, “Two-dimensional aperture synthesis in lunar radar astronomy,” Proc. R. Soc. London Ser. A 303, 477–491 (1968).
[CrossRef]

Radon, J.

J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integrralwerte langs gewisser Mannigfaltigkeiten,” Ber. Verhb. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262–277 (1917).

Saunders, R. S.

R. F. Jurgens, M. A. Slade, R. S. Saunders, “Evidence for highly reflecting materials on the surface and subsurface of Venus,” Science 240, 1021–1023 (1988).
[CrossRef] [PubMed]

Slade, M. A.

J. K. Harmon, M. A. Slade, “Radar mapping of Mercury: full-disk images and polar anomalies,” Science 258, 640–643 (1992).
[CrossRef] [PubMed]

D. O. Muhleman, B. J. Butler, A. W. Grossman, M. A. Slade, “Radar images of Mars,” Science 253, 1508–1513 (1991).
[CrossRef] [PubMed]

R. F. Jurgens, M. A. Slade, R. S. Saunders, “Evidence for highly reflecting materials on the surface and subsurface of Venus,” Science 240, 1021–1023 (1988).
[CrossRef] [PubMed]

Stacy, N. J. S.

N. J. S. Stacy, “High resolution synthetic aperture radar observations of the Moon,” Ph.D dissertation (Cornell University, Ithaca, N.Y., 1993).

Taylor, J. H.

J. H. Taylor, “Two-dimensional brightness distributions of radio sources from Lunar occultation observations,” Astrophysics 150, 421–426 (1967).
[CrossRef]

Thomson, J. H.

J. H. Thomson, J. E. B. Ponsonby, “Two-dimensional aperture synthesis in lunar radar astronomy,” Proc. R. Soc. London Ser. A 303, 477–491 (1968).
[CrossRef]

Vangenderen, J. L.

R. Gens, J. L. Vangenderen, “SAR interferometry—issues, techniques, applications,” Int. J. Remote Sensing 17, 1803–1835 (1996).
[CrossRef]

Astron. J. (1)

D. O. Muhleman, “Radar scattering from Venus and the Moon,” Astron. J. 69, 34–41 (1964).
[CrossRef]

Astrophysics (1)

J. H. Taylor, “Two-dimensional brightness distributions of radio sources from Lunar occultation observations,” Astrophysics 150, 421–426 (1967).
[CrossRef]

Ber. Verhb. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. (1)

J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integrralwerte langs gewisser Mannigfaltigkeiten,” Ber. Verhb. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262–277 (1917).

Geophysics (1)

T. S. Durrani, D. Bisset, “The Radon transform and its properties,” Geophysics 49, 1180–1187 (1984).
[CrossRef]

Int. J. Remote Sensing (1)

R. Gens, J. L. Vangenderen, “SAR interferometry—issues, techniques, applications,” Int. J. Remote Sensing 17, 1803–1835 (1996).
[CrossRef]

Inverse Probl. (1)

L. A. Kunyansky, “Generalized and attenuated Radon transforms: restorative approach to the numerical inversion,” Inverse Probl. 8, 809–819 (1992).
[CrossRef]

Proc. IEEE (1)

W. T. K. Johnson, “Magellan imaging radar mission to Venus,” Proc. IEEE 79, 777–790 (1991).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. H. Thomson, J. E. B. Ponsonby, “Two-dimensional aperture synthesis in lunar radar astronomy,” Proc. R. Soc. London Ser. A 303, 477–491 (1968).
[CrossRef]

Rev. Mod. Phys. (1)

S. J. Ostro, “Planetary radar astronomy,” Rev. Mod. Phys. 65, 1235–1279 (1993).
[CrossRef]

Science (3)

J. K. Harmon, M. A. Slade, “Radar mapping of Mercury: full-disk images and polar anomalies,” Science 258, 640–643 (1992).
[CrossRef] [PubMed]

R. F. Jurgens, M. A. Slade, R. S. Saunders, “Evidence for highly reflecting materials on the surface and subsurface of Venus,” Science 240, 1021–1023 (1988).
[CrossRef] [PubMed]

D. O. Muhleman, B. J. Butler, A. W. Grossman, M. A. Slade, “Radar images of Mars,” Science 253, 1508–1513 (1991).
[CrossRef] [PubMed]

SIAM Rev. (1)

Y. Nievergelt, “Elementary inversion of Radon’s transform,” SIAM Rev. 28, 79–84 (1986).
[CrossRef]

Other (2)

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

N. J. S. Stacy, “High resolution synthetic aperture radar observations of the Moon,” Ph.D dissertation (Cornell University, Ithaca, N.Y., 1993).

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Figures (5)

Fig. 1
Fig. 1

Geometry of the method. The spacecraft is located at O, directly above the subspacecraft point S. The point C is the pole of the planet, and the radar beam axis is along OC. The solid ellipse is the beam footprint. The dotted curves are lines of latitude (although the polar region is approximated here as a plane), and the dashed curves are the contours of constant Doppler shift. The spacecraft velocity is given by the vectorv.

Fig. 2
Fig. 2

Results of inverting the simulated data for OC: (a) the target surface, (b) the OC weighting function, (c) the result of inversion before correcting for the weighting function, and (d) the result of inversion after dividing out the weighting function.

Fig. 3
Fig. 3

Results of inverting the simulated data for SC: (a) the target surface, (b) the SC weighting function, (c) the result of inversion before correcting for the weighting function and (d) the result of inversion after dividing out the weighting function.

Fig. 4
Fig. 4

Ratio of surface reflectivity in the OC and SC: (a) the ratio for the target surfaces and (b) the ratio inferred from the inversions of simulated data. In both cases the maps have been scaled so that only values of μSCOC greater than 0.8 are shown.

Fig. 5
Fig. 5

Results of a resolution test: (a) The central 900 km2 of a target surface designed to test resolution and (b) the result of inversion and beam correction. This result indicates that the resolution of the method is approximately 1 km for the mission parameters used (orbital altitude of 150 ± 5 km, nadir pointing with a 3.2° standard deviation in orientation, radar power of 10 W, and a noise temperature of 1000 K).

Equations (32)

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vd=-v·rr.
vd=-xvx2+y2+H21/2.
δν=-2vdv0c,
y2=x2v2/vd2-1-H2,
Y2=X2/V2-1,
V=v2/vd2-1-1/2.
X=V cosh γ,
Y=sinh γ.
dXdY=cosh2 γdVdγ.
ΔPV=VV+ΔVdV -+dγμX,YWX,YH2 cosh2 γ,
dV=v2vd3-1-3/2v2vd3dvd=V3v2vd3dvd.
dPr=PtϕAe2μcosθFθλ2R4dA,
R2=x2+y2+H2,
cosθ=H/R,
cosϕ=xx0+yy0+H2Rx02+y02+H21/2,
Fθ=K1α cos θsin θ+α cos θ3.
Fθ=32πcos2 θ.
WX,Y=PtϕAe2FθcosθR4λ2,
ΔPvd,α=VV+ΔVdU-+dγWX,YTαμX,Y,
ΔxH2Δνν0cv.
gx,y=limq01π0πdα-+dp×gp+x cos α+y sin α, αGqp,
Gqp=1πq2p q1πq21-1-q2/p2-1/2pq.
gx,y=-12π··D·g,
gx, y=limq  01π0πdα -+dp×gpx cos α+y sin αy cos α-x sin α2+11/2,αGqp.
LX,Y=limq01π0πdα -+dv×ΔPV+X cos α+Y sin αY cos α-X sin α2+11/2, α×GqV,
LX,Y=μX,YWX,Y.
LX,Y=1Mi=1Mj=-Nj=+N ΔVjΔPijGq×Vj+X cos iΔα+Y sin iΔαY cos iΔα-X sin iΔα2+11/2,
bandwidth=4ν0vθmaxc1+θmax21/2,
k+1X, Y =DX, Y+-1-WkX,Y,
Wr,θ=SrQr,θ.
Sr=W2W,
PN=kBTRΔν,

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