Abstract

The many properties of radar echoes and other radiative systems were recently described by Gabriel [J. Opt. Soc. Am. A 19, 946 (2002)] as lower-dimensional projections of simple forms in special relativity. A broader treatment including coherent phenomena is summarized, in which the phase properties of radar images and interferograms are also shown to have a simple unified structure. Their apparent complexity is a result of projection onto the lower dimension(s) of the observation. A predicted new property, locally scalable (affine) phase, is observed in a radar interferogram.

© 2004 Optical Society of America

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References

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  1. A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, J. Geophys. Res. 94, 9183 (1989).
    [CrossRef]
  2. G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).
  3. A. K. Gabriel, J. Opt. Soc. Am. A 19, 946 (2002).
    [CrossRef]
  4. E. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, San Francisco, Calif., 1999).
  5. A. K. Gabriel, “Fundamental radar properties II: coherent phenomena in spacetime,” submitted to J. Opt. Soc. Am. A.
  6. A. K. Gabriel, IEEE Trans. Geosci. Remote Sensing 40, 1885 (2002).
    [CrossRef]
  7. A. K. Gabriel and R. M. Goldstein, Int. J. Remote Sensing 9, 857 (1988).
    [CrossRef]
  8. F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
    [CrossRef]

2002 (2)

A. K. Gabriel, IEEE Trans. Geosci. Remote Sensing 40, 1885 (2002).
[CrossRef]

A. K. Gabriel, J. Opt. Soc. Am. A 19, 946 (2002).
[CrossRef]

1994 (1)

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
[CrossRef]

1989 (1)

A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, J. Geophys. Res. 94, 9183 (1989).
[CrossRef]

1988 (1)

A. K. Gabriel and R. M. Goldstein, Int. J. Remote Sensing 9, 857 (1988).
[CrossRef]

Franceschetti, G.

G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

Gabriel, A. K.

A. K. Gabriel, IEEE Trans. Geosci. Remote Sensing 40, 1885 (2002).
[CrossRef]

A. K. Gabriel, J. Opt. Soc. Am. A 19, 946 (2002).
[CrossRef]

A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, J. Geophys. Res. 94, 9183 (1989).
[CrossRef]

A. K. Gabriel and R. M. Goldstein, Int. J. Remote Sensing 9, 857 (1988).
[CrossRef]

A. K. Gabriel, “Fundamental radar properties II: coherent phenomena in spacetime,” submitted to J. Opt. Soc. Am. A.

Gatelli, F.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
[CrossRef]

Goldstein, R. M.

A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, J. Geophys. Res. 94, 9183 (1989).
[CrossRef]

A. K. Gabriel and R. M. Goldstein, Int. J. Remote Sensing 9, 857 (1988).
[CrossRef]

Lanari, R.

G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

Monti-Guarnieri, A.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
[CrossRef]

Parizzi, F.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
[CrossRef]

Pasquali, P.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
[CrossRef]

Prati, C.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
[CrossRef]

Rocca, F.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
[CrossRef]

Taylor, E.

E. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, San Francisco, Calif., 1999).

Wheeler, J. A.

E. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, San Francisco, Calif., 1999).

Zebker, H. A.

A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, J. Geophys. Res. 94, 9183 (1989).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (2)

A. K. Gabriel, IEEE Trans. Geosci. Remote Sensing 40, 1885 (2002).
[CrossRef]

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, IEEE Trans. Geosci. Remote Sensing 32, 855 (1994).
[CrossRef]

Int. J. Remote Sensing (1)

A. K. Gabriel and R. M. Goldstein, Int. J. Remote Sensing 9, 857 (1988).
[CrossRef]

J. Geophys. Res. (1)

A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, J. Geophys. Res. 94, 9183 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Other (3)

G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

E. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, San Francisco, Calif., 1999).

A. K. Gabriel, “Fundamental radar properties II: coherent phenomena in spacetime,” submitted to J. Opt. Soc. Am. A.

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Figures (4)

Fig. 1
Fig. 1

Light cones in 3-D spacetime. The cones of transmitter T and receiver R are separated (outermost vertices) by some time τc. The illuminated regions of the plane are aTx,t for T and aRx,t for R. Radiation from a T cone can scatter onto an R cone and be detected t>τc/2 later. The diamond of revolution is the region where causality permits T to communicate with R.

Fig. 2
Fig. 2

Light cones and phases in the x,t plane. The T pulse worldlines, aT, increase in t (four phase cycles). Phase velocity (slope), which is infinite at nadir, approaches c for large x. R worldlines, aR, decrease by choice (four cycles). The first echo returns at the offset τc=2z0/c. The diamond-shaped region is where T can communicate with R.

Fig. 3
Fig. 3

Light cones and scene plane in x,y,t. The cones are figures of rotation of the a*x,t (t is now vertical). The height z0 causes the cone curvature. The scene x,y is orthogonal to the t axis; an independent x length Bx causes a rotation.

Fig. 4
Fig. 4

Affine phase observation. A radar interferogram (monochrome rendering) with diverging orbits is shown. The linear region is where spacetime curvature vanishes.

Equations (2)

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Γx,τc=-aTx,taRx,t-τcdt,
λ/4π2ϕ-ϕ0-2πfxx2=Bz2+Bx2,

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