Relativistic Corrections to Spaceborne SAR Data

1. Relativistic Corrections to Spaceborne SAR Data K. Jezek April 7, 2009

2. Classical Doppler Shift • Doppler frequency is used to determine the azimuth position of a point on the surface r x q

3. Doppler shift measured by detector coincident with transmitter(relativisticaly correct given constancy of the speed of light!) cDt” ‘’ w”Dt” cDt” N. Ashby

4. Writing Dt as 2p/(kc) (seconds per cycle) • Here fsis the operating frequency of the source, va is the propagation velocity of the wave, and vs is the relative speed of the source as viewed along the line connecting source to target (vsin(q)).

5. The stationary receiver/target now scatters signal with frequency fd back to the source. The signal observed by the moving source is • Now evaluate the difference between the transmitter frequency and the observed Doppler frequency. • Curlander and McDonough (1991, p. 17) • Note that when theta = 0, the transmitted signal is not doppler shifted

6. Including Earth rotation • Where vt is the speed of the satellite, veeq is the speed of the earth at the equator, a is the orbit inclination and c is the speed of light • Note that when theta is zero, the signal is doppler shifted (squinted). The consequence is an azimuthal bias if earth rotation is not taken into account. • Are there other factors that can cause squint and might not be accounted for?

7. Moving Targets • Train moving at 40 km/hr and displaced about 800 m from track Uwe Stilla

8. Relativistic Doppler • A stationary clock appears to run faster relative to a clock that is moving. Consequently there is a relativistic correction to the doppler shift given by • Where theta is measured in the frame of the transmitter

9. Neglecting earth rotation, I play the same game for a signal transmitted, scattered and received. I estimate that • Seems to be consistent with Ashby eq 103! • When theta is 0 degrees, the satellite velocity is orthogonal to the line connecting the satellite position and the target. For the relativistic case, the Doppler value is • When theta is zero, the relativistic doppler is not zero! Consequently, time dilation introduces a small amount of squint

10. Comparison between relativistic and classical estimates of the difference between the transmitter frequency and the Doppler frequency. The x-axis is the angle between the satellite velocity vector and the line connecting the satellite to the target (90 degrees corresponds to the classical zero-doppler case). Satellite speed is 7 km/sec. Transmitter frequency is 5 GHz.

11. Examine the squint bias by expanding terms in the doppler equation Neglecting higher order terms

12. The squint bias can be estimated by expressing the sine of the angle in terms of the azimuth offset and range • Notice that this reduces to the classical result when B is ignored.

13. Consider the broadside case again. Suppose we do not use the proper biased Doppler and simply take fd’=fs • Then x will be in error by an amount. • This corresponds to about a 10 m bias for 1000 km range (about half that estimated by Olmstead). • Now the squint correction is in fact tiny in terms of frequency. It is observable at a macroscopic scale because of the long lever arm acting on the small azimuth angle between satellite and the target on the surface of the Earth.

14. A warning • The consequence of time dilation can be regarded as either a relativistic doppler shift or aberration (the apparent change in propagation direction as viewed by a moving or a stationary observer). These are essentially the time domain and frequency expressions of the effect. I am not sure I have properly accounted for aberration in my analysis (should there be an additional correction on theta?). However, the fact that my expressions reduce to the classical results gives me some confidence. Also my result is the same as Ashby eq. 103.

15. To consider • The effect seems to be rediscovered from time to time. Is there a way to make a test of this prediction? What sources of error might confound an experiment? • How do the relativistic doppler equations change when earth rotation is included? • The discussion is in terms of a point target. Does this analysis translate to a distributed target? • What might be the consequences for systems like Cryosat? • See the Ashby paper for an interesting prediction on the offsetting effects of time dilation red shifts and gravitation blue shifts. Is there a sweet spot for orbital elevations where the timing corrections cancel out?

16. A strange diversion • Faraday’s empirical law is; • In virtually all texts, this is used to state Maxwell’s curl equation as

17. The switcheroo from total to partial derivative is crucial because • it establishes a basis for the constancy of the speed of light • it establishes spacetime symmetry by balancing the equations as • Were the full material derivative retained, then there would be an addition all dot product (as originally proposed by Hertz)

18. Now the Hertz view (which included the ether postulate) was discarded after experiments like Michelson and Morley. • Clearly, observation should dictate notation. But then there seems to be a conflict between the Faraday and M&M experiments at least notation wise. Moreover, why does the present math, which leads to SRT, seem so clumsy?

19. Phipps in his privately published book “Old Physics for New” argues that Hertz was correct in his math but wrong in his interpretation of the symbols. Rather than the material derivative being related to the ether velocity, the additional dot product term should describe the motion of the detector through the field. • Invoking Potiers principle (light is convected along by the physical medium, which could be a detector, to first order), Phipps goes on to compute a modified form of the doppler correction. Could it be testable with SAR? • I am not sure I believe it! But the slight of hand with derivatives is disturbing.