Abstract

A derivation of the properties of pulsed radiative imaging systems is presented with examples drawn from conventional, synthetic aperture, and interferometric radar. A geometric construction of the space and time components of a radar observation yields a simple underlying structural equivalence among many of the properties of radar, including resolution, range ambiguity, azimuth aliasing, signal strength, speckle, layover, Doppler shifts, obliquity and slant range resolution, finite antenna size, atmospheric delays, and beam- and pulse-limited configurations. The same simple structure is shown to account for many interferometric properties of radar: height resolution, image decorrelation, surface velocity detection, and surface deformation measurement. What emerges is a simple, unified description of the complex phenomena of radar observations. The formulation comes from fundamental physical concepts in relativistic field theory, of which the essential elements are presented. In the terminology of physics, radar properties are projections of hidden variables—curved worldlines from a broken symmetry in Minkowski space–time—onto a time-serial receiver.

© 2002 Optical Society of America

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References

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  1. C. Elachi, Spaceborne Radar Remote Sensing: Applications and Techniques (Wiley, New York, 1987).
  2. E. Lieb, “Integral bounds for radar ambiguity functions and Wigner distributions,” J. Math. Phys. 31, 594–599 (1990).
    [CrossRef]
  3. G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC, Baton Rouge, La., 1999).
  4. R. Altes, E. Titlebaum, “Graphical derivations of radar, sonar, and communication signals,” IEEE Trans. Aerosp. Electron. Syst. AES-11, 38–44 (1975).
    [CrossRef]
  5. E. Taylor, J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, New York, 1999).
  6. A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing (Prentice–Hall, Englewood Cliffs, N.J., 1999).
  7. T. Gilliam, Time Bandits (Handmade Films, London, 1981).
  8. H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
    [CrossRef]
  9. R. M. Goldstein, T. P. Barnett, H. A. Zebker, “Remote sensing of ocean currents,” Science 246, 1282–1285 (1989).
    [CrossRef] [PubMed]
  10. A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
    [CrossRef]

1990 (1)

E. Lieb, “Integral bounds for radar ambiguity functions and Wigner distributions,” J. Math. Phys. 31, 594–599 (1990).
[CrossRef]

1989 (2)

R. M. Goldstein, T. P. Barnett, H. A. Zebker, “Remote sensing of ocean currents,” Science 246, 1282–1285 (1989).
[CrossRef] [PubMed]

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

1986 (1)

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

1975 (1)

R. Altes, E. Titlebaum, “Graphical derivations of radar, sonar, and communication signals,” IEEE Trans. Aerosp. Electron. Syst. AES-11, 38–44 (1975).
[CrossRef]

Altes, R.

R. Altes, E. Titlebaum, “Graphical derivations of radar, sonar, and communication signals,” IEEE Trans. Aerosp. Electron. Syst. AES-11, 38–44 (1975).
[CrossRef]

Barnett, T. P.

R. M. Goldstein, T. P. Barnett, H. A. Zebker, “Remote sensing of ocean currents,” Science 246, 1282–1285 (1989).
[CrossRef] [PubMed]

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing (Prentice–Hall, Englewood Cliffs, N.J., 1999).

Elachi, C.

C. Elachi, Spaceborne Radar Remote Sensing: Applications and Techniques (Wiley, New York, 1987).

Franceschetti, G.

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC, Baton Rouge, La., 1999).

Gabriel, A. K.

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

Gilliam, T.

T. Gilliam, Time Bandits (Handmade Films, London, 1981).

Goldstein, R. M.

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

R. M. Goldstein, T. P. Barnett, H. A. Zebker, “Remote sensing of ocean currents,” Science 246, 1282–1285 (1989).
[CrossRef] [PubMed]

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Lanari, R.

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC, Baton Rouge, La., 1999).

Lieb, E.

E. Lieb, “Integral bounds for radar ambiguity functions and Wigner distributions,” J. Math. Phys. 31, 594–599 (1990).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing (Prentice–Hall, Englewood Cliffs, N.J., 1999).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing (Prentice–Hall, Englewood Cliffs, N.J., 1999).

Taylor, E.

E. Taylor, J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, New York, 1999).

Titlebaum, E.

R. Altes, E. Titlebaum, “Graphical derivations of radar, sonar, and communication signals,” IEEE Trans. Aerosp. Electron. Syst. AES-11, 38–44 (1975).
[CrossRef]

Wheeler, J. A.

E. Taylor, J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, New York, 1999).

Zebker, H. A.

R. M. Goldstein, T. P. Barnett, H. A. Zebker, “Remote sensing of ocean currents,” Science 246, 1282–1285 (1989).
[CrossRef] [PubMed]

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

R. Altes, E. Titlebaum, “Graphical derivations of radar, sonar, and communication signals,” IEEE Trans. Aerosp. Electron. Syst. AES-11, 38–44 (1975).
[CrossRef]

J. Geophys. Res. (2)

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

J. Math. Phys. (1)

E. Lieb, “Integral bounds for radar ambiguity functions and Wigner distributions,” J. Math. Phys. 31, 594–599 (1990).
[CrossRef]

Science (1)

R. M. Goldstein, T. P. Barnett, H. A. Zebker, “Remote sensing of ocean currents,” Science 246, 1282–1285 (1989).
[CrossRef] [PubMed]

Other (5)

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC, Baton Rouge, La., 1999).

E. Taylor, J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, New York, 1999).

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing (Prentice–Hall, Englewood Cliffs, N.J., 1999).

T. Gilliam, Time Bandits (Handmade Films, London, 1981).

C. Elachi, Spaceborne Radar Remote Sensing: Applications and Techniques (Wiley, New York, 1987).

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Figures (16)

Fig. 1
Fig. 1

A light cone in Minkowski space originating at some arbitrary height z0 above the x, t plane forms a hyperbola where it intersects the plane. Note that the plane in the figure contains only one scene spatial dimension, the distance x.

Fig. 2
Fig. 2

Along-ground distance of incoming and outgoing radar window worldlines aT and aR for a δ-function pulse.

Fig. 3
Fig. 3

Intersecting light cones of the transmitter and receiver. The cone vertices at height z0 above the (x, t) plane are separated in time by τ0, the pulse length, for like-looking cone pairs and by τ, the convolution offset, for opposite-looking pairs measured from the outermost vertices. The hyperbolas in the (x, ct) or (distance, time) plane in the figure also intersect, forming the image resolution element (rezel) shown as the diamond shape.

Fig. 4
Fig. 4

Spherical pulse radii aT(t) and aT(t-τ0) on the ground as a function of time for a transmitter at 800 km altitude with exaggerated pulse length z0/3c. The temporal (horizontal) separation between the lines is τ0 at all ranges x, usually the inverse of the transmitter (or receiver) bandwidth.

Fig. 5
Fig. 5

Radial pulse width as a function of time.

Fig. 6
Fig. 6

Components of the convolution Γ that determine the receiver response of a point target for z0=800 km and an exaggerated pulse width z0/3c=τ0. The transmitter rect( ) window worldlines aT are the two traces rising to the right; the receiver rect( ) window worldlines aR fall to the right. The offset variable τ is measured along the time axis as the distance between the outermost two traces. The diamond-shaped area is the space–time channel ζ; the horizontal line is the worldline of a target at some distance x1.

Fig. 7
Fig. 7

A real overlap of the integration of Eq. (2) as a function of the offset τ, scaled to show the space–time channel. Horizontal axes are in units of τ0, vertical axes in units of z0. (a) Near the origin (x=0), where τ=0; (b) in the intermediate region, where τ is a low multiple of t0; (c) close-up of the far range, τt0. Each plot contains a finite-pulse aT and aR; the space–time channel is the light area where both functions are nonzero; the dark areas are where one or the other function, but not both, is nonzero; the remaining background areas are where both functions are zero. Only the light area contributes to the integral Γ.

Fig. 8
Fig. 8

Numerical size of space–time channel (ambiguity) region for uniform scatterers for a monostatic SAR at altitude 800 km with exaggerated pulse width τ0=z0/3c as a function of separation τ.

Fig. 9
Fig. 9

A rotation γ can be viewed as defining a new radar height z0z0, which changes the imaging properties.

Fig. 10
Fig. 10

Bistatic outgoing-pulse worldline aT for a transmitter on a spacecraft and time-reversed or incoming window aR for a receiver at a lower altitude for a very short pulse. The two curvatures are different because the receiver height z0 is 50 km versus 800 km for the transmitter [Eq. (1)].

Fig. 11
Fig. 11

Image plane (x, y) (distances in meters; x and y are scaled to z0). Finite-pulse aT and aR intersect with the plane (x, y) for a bistatic configuration (either ring can be T or R, with the other then R or T). The bistatic subradar points differ, accounting for the spatial offset of the rings. The relative sizes of the rings are determined by the offset τ (see Fig. 3).

Fig. 12
Fig. 12

Bistatic worldlines aT and aR calculated from Eq. (1) for the case of imaging of the area between a separated transmitter and receiver. The lowest line is aT, here offset in the -x direction by some amount that reflects the distance from the transmitter’s nadir to the receiver’s nadir (lower square-root branch is suppressed); the upper two connected lines are the two square-root branches of aR (both branches of the intersection of the incoming aR light cone with the (x, t) plane). A slight relative shift (change in τ) from what is in the figure would create a huge space–time channel (the two lower lines would coincide over a large space–time area).

Fig. 13
Fig. 13

Ellipsoidal ambiguity surfaces for bistatic radar in the (x, z) plane. The T and R antennas are the foci of the ellipses; the lower horizontal line is the x dimension of the scene (the ground).

Fig. 14
Fig. 14

Schematic representation of reception; two incoming radiation worldlines aR framing a flat-scene rezel (i.e., separated temporally by the inverse of the transmitter bandwidth) are shown with the model of the receiver temporal aperture. The receiver integrates across the interval defined by the same bandwidth as the pulse; the receiver phase associated with the indicated rezel is definitionally measured when the receiver is synchronized with and integrating the indicated interval.

Fig. 15
Fig. 15

Worldlines associated with azimuth aliasing by receiver. The circles indicate the phase measurements ϕ1, ϕ2, ϕ3 made at discrete sampling times. The third measurement is assumed to include a subsampling error of 2π, causing the receiver to produce a false target in the image, which can be thought of as a ghost worldline in space–time.

Fig. 16
Fig. 16

Worldline pairs for interferometric radar. Each pair cT* is composed of aT(t) and aT(t-τ); cR* is composed of aR(t) and aR(t-τ).

Equations (7)

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x(t)=(ct)2-z02.
Γ(τ)=-δ(t)δ(t-τ)dt.
Γ(τ)=Γ(x, τ)=-aT(x, t)aR(x, (t-τ))dt,
ζ(x, t, τ)aT(x, t)aR(x, (t-τ)).
Γ(τ)-aT(x0-ct)aR(x0-c(t-τ0))dt,
ζt(t, τ)rect(xT(t)/cτ0)rect(xR(t-τ)/cτ0)σ(xt),
Γ(τ)=σ tri((cτ)/2cτ0) forτ>t0,

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