Abstract

A previous publication [J. Opt. Soc. Am. A 19, 946–956 (2002) ] presented a general formulation of radiative systems based on special relativity, and properties of imaging radar were derived as examples. Complex and diverse properties of radar images were shown to have a simple and unified origin when viewed as lower-dimensional (temporal) projections of the space–time structure of a radar observation. A diagram was developed that could be manipulated for a simple, intuitive view of the underlying structure of radar observations and phenomena. That treatment is here extended to include coherent phenomena as they appear in the lower time dimensions of the image. Various known coherent properties of imaging radar and interferometry are derived. The formulation is shown to be a generalization of a conventional echo correlation and is extended to a second spatial dimension. From this perspective, coherent properties also have a surprisingly simple and unified structure; their observed complexity is somewhat illusory, also a consequence of projection onto the lower temporal dimension of the receiver. While this formulation and the rules governing it are quite different from the standard treatments, they have the considerable advantage of providing a much simpler, intuitive, and unified description of radiative (radar and optical) systems that is rooted in fundamental physics.

© 2008 Optical Society of America

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References

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  1. A. K. Gabriel, "Fundamental radar properties: hidden variables in space-time," J. Opt. Soc. Am. A 19, 946-956 (2002).
    [CrossRef]
  2. G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing (CRC Press, 1999).
  3. E. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, 1999).
  4. A. K. Gabriel, "Unification of radar phenomena as spacetime curvature: prediction and observation of affine-phase effect," Opt. Lett. 29, 1533-1555 (2004).
    [CrossRef] [PubMed]
  5. A. K. Gabriel and R. M. Goldstein, "Crossed orbit interferometry: theory and experimental results from SIR-B," Int. J. Remote Sens. 9, 857-872 (1988).
    [CrossRef]
  6. A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, "Mapping small elevation changes over large areas: Differential radar interferometry," J. Geophys. Res. 94, 9183-9191 (1989).
    [CrossRef]
  7. A. K. Gabriel, "A simple model for SAR azimuth speckle, focusing and interferometric correlation," IEEE Trans. Geosci. Remote Sens. 40, 1885-1888 (2002).
    [CrossRef]
  8. F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
    [CrossRef]
  9. T. Gilliam, Time Bandits (Handmade Films, 1981).

2004 (1)

2002 (2)

A. K. Gabriel, "Fundamental radar properties: hidden variables in space-time," J. Opt. Soc. Am. A 19, 946-956 (2002).
[CrossRef]

A. K. Gabriel, "A simple model for SAR azimuth speckle, focusing and interferometric correlation," IEEE Trans. Geosci. Remote Sens. 40, 1885-1888 (2002).
[CrossRef]

1999 (2)

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

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

1994 (1)

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
[CrossRef]

1989 (1)

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

1988 (1)

A. K. Gabriel and R. M. Goldstein, "Crossed orbit interferometry: theory and experimental results from SIR-B," Int. J. Remote Sens. 9, 857-872 (1988).
[CrossRef]

1981 (1)

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

Franceschetti, G.

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

Gabriel, A. K.

A. K. Gabriel, "Unification of radar phenomena as spacetime curvature: prediction and observation of affine-phase effect," Opt. Lett. 29, 1533-1555 (2004).
[CrossRef] [PubMed]

A. K. Gabriel, "A simple model for SAR azimuth speckle, focusing and interferometric correlation," IEEE Trans. Geosci. Remote Sens. 40, 1885-1888 (2002).
[CrossRef]

A. K. Gabriel, "Fundamental radar properties: hidden variables in space-time," J. Opt. Soc. Am. A 19, 946-956 (2002).
[CrossRef]

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

A. K. Gabriel and R. M. Goldstein, "Crossed orbit interferometry: theory and experimental results from SIR-B," Int. J. Remote Sens. 9, 857-872 (1988).
[CrossRef]

Gatelli, F.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
[CrossRef]

Gilliam, T.

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

Goldstein, R. M.

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

A. K. Gabriel and R. M. Goldstein, "Crossed orbit interferometry: theory and experimental results from SIR-B," Int. J. Remote Sens. 9, 857-872 (1988).
[CrossRef]

Lanari, R.

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

Monti-Guarnieri, A.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
[CrossRef]

Parizzi, F.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
[CrossRef]

Pasquali, P.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
[CrossRef]

Prati, C.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
[CrossRef]

Rocca, F.

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
[CrossRef]

Taylor, E.

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

Wheeler, J. A.

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

Zebker, H. A.

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

IEEE Trans. Geosci. Remote Sens. (2)

A. K. Gabriel, "A simple model for SAR azimuth speckle, focusing and interferometric correlation," IEEE Trans. Geosci. Remote Sens. 40, 1885-1888 (2002).
[CrossRef]

F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, "The wavenumber shift in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 32, 855-864 (1994).
[CrossRef]

Int. J. Remote Sens. (1)

A. K. Gabriel and R. M. Goldstein, "Crossed orbit interferometry: theory and experimental results from SIR-B," Int. J. Remote Sens. 9, 857-872 (1988).
[CrossRef]

J. Geophys. Res. (1)

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

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

Opt. Lett. (1)

Other (3)

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

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

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

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Figures (14)

Fig. 1
Fig. 1

Light cones representing the space–time regions of an outgoing radar pulse and an incoming receiver window. As observed along the antenna worldline, the pulse starts at the leftmost vertex and stops at the vertex τ 0 later; similar comments apply to the incoming window a R ( x , t ) . The pulse length τ 0 is then the distance along the time axis between the like-oriented cone vertices. The cone intersections with the ( x , t ) plane are the hyperbolas a T ( x , t ) and a R ( x , t ) . The small diamond-shaped window is the region where causality allows the transmitter and receiver to communicate.

Fig. 2
Fig. 2

Space–time components of radar light cones in the ( x , t ) plane. The transmitter pulse worldlines a T ( x , t ) are the two lines rising from left to right, representing the apparently superluminal motion on the ground of a transmitted pulse of length τ 0 . The phase velocity c is infinite at nadir and is asymptotic to c for large x. Similarly, the receiver worldlines a R ( x , t ) fall from left to right. The offset variable τ is the interval along the time axis between the outermost two traces, equal to 2 z 0 c when the emitted pulse first returns to the receiver. The diamond-shaped area is the spacetime channel, and the horizontal line is the worldline of a stationery target at some distance x t .

Fig. 3
Fig. 3

Visualization of phase as an extra dimension of space–time (phase can be measured only along x = 0 ). The transmitter phase is chosen to increase in time; the receiver phase is chose to decrease. The pitched line segments beneath the ( x , t ) plane are the phases of the outgoing and incoming pulses (temporal length τ 0 ) referenced to zero at the phase axis origin, which was chosen as the halfway time between transmission and reception (the convention for the phase origin is usually ϕ 0 = 0 at the antenna center with unspecified temporal origin).

Fig. 4
Fig. 4

Visualization of phase as an extra dimension of space–time (rotated for better perspective). The transmitter phase is represented as an increasing linear ramp imposed on the outgoing a T ( x , t ) ; the receiver phase is represented as a conjugate (decreasing) ramp imposed on the incoming a R ( x , t ) . The intersection area shown in the far field is the causal channel. The combined (heterodyne) phase inside the channel increases with distance but does not change in time, as is expected for a stationary target.

Fig. 5
Fig. 5

Space time rezel in the far field with transmitter and receiver phases visualized as an extra dimension; within the channel the phases are added to represent the removal of the carrier frequency. The phase of a target (measured at the receiver) is the result of the temporal integration over the channel [1].

Fig. 6
Fig. 6

Transmitter and receiver phases visualized as an extra dimension in space–time for the nadir (closest) rezel.

Fig. 7
Fig. 7

Sawtooth phase representing 2 π (here [0,1]) phase wrapping with four cycles of phase.

Fig. 8
Fig. 8

Shaded visualization of phase wrapping on a T ( x , t ) and a R ( x , t ) worldlines (phase dimension perpendicular to the page, shown as shading). As in the previous plots, the outgoing phase is presented as having the opposite slope of the incoming phase in order to visualize carrier removal.

Fig. 9
Fig. 9

Space–time visualization of wrapped phases at nadir, where the worldline a * ( x , t ) both increases the causal channel (and, so, x rezel size) and creates a spatial phase chirp, visible from the isophase contours (shade) in the channel, which are not evenly spaced in the x direction (the tick marks drawn to the right are placed at each full cycle of phase in x; they are not evenly spaced in x).

Fig. 10
Fig. 10

Perspective view of the annulus-shaped worldline functions a T ( x , y , t ) and a R ( x , y , t ) for some t as they appear in the scene plane ( x , y ) . The a T ( x , y , t ) ring expands in time, while the a R ( x , y , t ) ring contracts; Fig. 5 of [1] shows the changing radial width of such a time-delimited annulus. When the rings overlap, they form a planar ( x , y ) slice of the causal channel which has two space dimensions, i.e., ς ( x , t ) and ς ( x , y , t ) .

Fig. 11
Fig. 11

Cutaway views of a T ( x , y , t ) and a R ( x , y , t ) , the generalizations of worldlines a T ( x , t ) and a R ( x , t ) to a second spatial dimension y (time is now vertical) for τ 0 = 0 (δ-function pulse). The axis of symmetry is the worldline of the transmitter (T) and receiver (R). Each panel represents a different value of length z 0 ; the resulting curvature changes in the a * ( x , y , t ) mean that the cones cannot be made identical by scaling.

Fig. 12
Fig. 12

Cutaway visualization of curved light cones a T ( x , y , t ) and a R ( x , y , t ) including finite pulse length τ 0 as the distance between like-facing cones. The causal channel is now the region of rotation of the small diamond shape (rezel).

Fig. 13
Fig. 13

Space–time phases of different observations made from locations x = 0 (black, A) and x = x 0 (gray, B). In this visualization, the causal channels of the two observations have been overlaid by shifting the B lines by x 0 relative to the A lines. The same location x t in the scene is observed at two different locations on the same isophases (and accordingly different causal channels from the same isophases). The different curvatures result in different phases across the rezel when observed along the t axis.

Fig. 14
Fig. 14

(Complementary diagram to Fig. 13) A scene is observed at some x t from two different altitudes z 0 (gray, B) and z 1 (black, A), resulting in two different curvatures for the worldlines a * ( x , y , t ) .

Tables (1)

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Table 1 Terms

Equations (30)

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Γ ( τ ) = Γ ( x ( t ) , τ ) = a T ( x , t ) a R ( x , ( t τ ) ) d t a T ( x , t ) τ a R ( x , t ) .
x ( t ) = ( c t ) 2 z 0 2 .
R ( τ ) = σ ( x ( t ) ) τ w ( t ) ,
Ω ( τ ) = σ ( x ( t ) ) τ w ( t ) τ w * ( t ) = σ ( x ( t ) ) τ I ( t ) ,
I ( τ ) = w ( t ) τ w * ( t ) .
x c t ,
Ω ( τ ) = σ ( c t ) τ I ( t ) ,
x ( t ) = c t ( 1 z 0 2 2 c 2 t 2 ) .
Ω ( τ ) = σ ( c t z 0 2 2 c t ) τ w ( t ) τ w * ( t ) .
t ( x ) = 1 c x 2 + z 0 2 ,
t ( x ) z 0 c ( 1 + x 2 2 z 0 2 ) = t 0 + x 2 2 c 2 z 0 2 .
Ω ( t ) = σ ( x ) τ w ( t ( x ) ) τ w * ( t ( x ) ) = σ ( x ) τ I ( τ ( x ) ) ,
Ω ( τ ) = σ ( x ) τ a T ( x , t ) τ 2 a R ( x , t ) = σ ( x ) τ Γ ( x , t ) ,
Ω ( τ τ 0 ) = σ ( x ) τ τ 0 a T ( x , t τ 0 ) τ 2 τ 0 a R ( x , t τ 0 ) = σ ( x ) τ τ 0 Γ ( x , t τ 0 ) .
ρ = x 2 + y 2 + z 0 2 ,
y f = ρ λ L ,
ρ λ 2 y f = L 2 .
d ϕ ( y = 0 ) = 2 π λ d ρ = 2 π λ [ ρ ( y = L 2 ) ρ ( y = 0 ) ] 2 π y 2 λ z 0 = 2 π ( L 2 ) 2 λ z 0 .
δ ( d ϕ ( y ) ) 2 π λ ( d ρ ( y ) d ρ ( y = 0 ) ) 4 π λ ( L 2 ) y z 0 .
y f = z 0 λ 2 ( L 2 ) .
a * ( x , y , t ) A = a * ( x , y , t ) ,
then the gray observation is described by
a * ( x , y , t ) B = a * ( x x 0 , y , t τ 2 ( x t , x 0 ) ) ,
d ϕ A = ϕ ( t 1 ) ϕ ( t 2 ) , d ϕ B = ϕ ( t 3 ) ϕ ( t 4 ) ,
δ ( d ϕ ) = d ϕ A d ϕ B = ϕ ( t 1 ) ϕ ( t 2 ) ϕ ( t 3 ) + ϕ ( t 4 ) .
r = x 2 + y 2 , ρ = r 2 + z 0 2 .
ρ d ρ = r d r = x d x + y d y .
d ρ = y ρ d y .
d ρ λ = 1 y = ρ λ ( d y ) π .
( d y ) π = ρ λ y .

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