Abstract

Frequency shifts of side-ranging lidar signals are calculated to high order in the small quantities (vc), where v is the velocity of a spacecraft carrying a lidar laser or of an aerosol particle that scatters the radiation back into a detector (c is the speed of light). Frequency shift measurements determine horizontal components of ground velocity of the scattering particle, but measured fractional frequency shifts are large because of the large velocities of the spacecraft and of the rotating earth. Subtractions of large terms cause a loss of significant digits and magnify the effect of relativistic corrections in determination of wind velocity. Spacecraft acceleration is also considered. Calculations are performed in an earth-centered inertial frame, and appropriate transformations are applied giving the velocities of scatterers relative to the ground.

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References

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  1. C. Møller, The Theory of Relativity, 2nd ed. (Clarendon Press, 1972) pp. 39-42.
  2. H. P. Robertson and T. W. Noonan, Relativity and Cosmology (Saunders, 1968) pp. 45 ff.
  3. R. D'Inverno, Introducing Einstein's Relativity (Clarendon Press, 1992) p. 40.
  4. V. S. R. Gudimetla and M. Kavaya, Special Relativity Corrections for Space-Based Lidar, Appl. Opt.38, 6374-6382 (1999).
  5. B. W. Parkinson and J. J. Spilker, Jr., Global Positioning System: Theory and Applications (American Institute of Aeronautics and Astronautics, 1996) Vol. I, pp. 161-165.

Other (5)

C. Møller, The Theory of Relativity, 2nd ed. (Clarendon Press, 1972) pp. 39-42.

H. P. Robertson and T. W. Noonan, Relativity and Cosmology (Saunders, 1968) pp. 45 ff.

R. D'Inverno, Introducing Einstein's Relativity (Clarendon Press, 1992) p. 40.

V. S. R. Gudimetla and M. Kavaya, Special Relativity Corrections for Space-Based Lidar, Appl. Opt.38, 6374-6382 (1999).

B. W. Parkinson and J. J. Spilker, Jr., Global Positioning System: Theory and Applications (American Institute of Aeronautics and Astronautics, 1996) Vol. I, pp. 161-165.

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Figures (9)

Fig. 1
Fig. 1

Earth-centered inertial coordinates. The origin is at earth’s center, and the x y z axes have fixed directions in space. For example, the x axis can point toward the vernal equinox with the z axis parallel to earth’s angular velocity vector. The positions of the scattering particle and the spacecraft are respectively denoted by r and r L in these coordinates.

Fig. 2
Fig. 2

Lidar rest frame. This is defined with origin at the lidar transmitter. Quantities measured in this frame such as the wind velocity w and the velocity V E of the ground point are doubly primed.

Fig. 3
Fig. 3

Earth reference frame. This is defined with origin at the ground point below the scatterer but fixed on earth’s surface. The wind velocity relative to the surface is the object of the lidar measurement and is denoted by w . Quantities such as the velocity V L of the spacecraft in this frame are denoted with primes.

Fig. 4
Fig. 4

Two wavefronts are shown traveling toward the scattering particle with speed c relative to the LRF. The particle has velocity w . The incident wavefronts are separated in space by the wavelength λ 0 . The figure is drawn in the LRF at the instant that wavefront #1 strikes the particle; the scatterer is sufficiently far from the transmitter that the arriving wavefronts are plane.

Fig. 5
Fig. 5

The first wavefront generates a spherical backscattered wave, shown at the instant the second wavefront strikes the scatterer. The backscattered wave becomes plane as it travels toward the detector. The scatterer moves with velocity w . Successive wavefronts striking the scatterer generate a series of backscattered waves separated in space by wavelength λ .

Fig. 6
Fig. 6

Possible situation with V L V E parallel to the ground, at angle ϕ from the uniform wind field. The scanning angle is 30° from the nadir.

Fig. 7
Fig. 7

Error in wind velocity resulting from neglect of denominator in last term of Eq. (41). The three curves are labeled by the scanning angle relative to the nadir.

Fig. 8
Fig. 8

Vectors for the simulation in Section 5, viewed from the ECI reference frame. Latitude–longitude grid lines are drawn at the time of the initial measurement 0.5° apart. Displacement of the ground point between the measurements is due to earth rotation and is along a parallel of latitude.

Fig. 9
Fig. 9

The relative velocities V L and V E are not antiparallel. The diagram is drawn in the plane formed by V L and V E .

Tables (1)

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Equations (159)

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λ 0 = c Δ t w cos θ Δ t .
λ = c Δ t + w cos θ Δ t .
λ 0 λ = c f 0 c f = c Δ t w Δ t cos θ c Δ t + w Δ t cos θ .
f f 0 = 1 w c cos θ 1 + w c cos θ = 1 w k c k 1 + w k c k .
f f 0 = 1 + Δ f f 0 ,
w k c k = 1 2 Δ f f 0 1 + 1 2 Δ f f 0 .
w n t = c 2 Δ f f 0 1 + 1 2 Δ f f 0 .
w n t = c 2 ( Δ f f 0 ) + c ( Δ f 2 f 0 ) 2 +
V E V E r V E 2 ,
γ E = 1 1 V E 2 c 2 .
r V E V E r V E 2 .
r = r V E V E r V E 2 + γ E ( V E V E r V E 2 + V E c c t ) ,
c t = γ E ( c t + V E r c ) ,
V E r = γ E ( V E r + V E 2 c c t ) .
r = r + ( γ E 1 ) V E V E r V E 2 γ E V E c c t ,
c t = γ E ( c t V E r c ) ,
V E r = γ E ( V E r V E 2 c c t ) .
d r = d r + ( γ E 1 ) V E V E d r V E 2 γ E V E c c d t ;
c d t = γ E ( c d t V E d r c ) ,
d r d t = v = d r + ( γ E 1 ) V E V E d r V E 2 γ E V E d t γ E ( d t V E d r c 2 ) = 1 V E 2 c 2 v + ( 1 1 V E 2 c 2 ) V E V E v V E 2 V E 1 V E v c 2 ,
v = 1 V E 2 c 2 v + ( 1 1 V E 2 c 2 ) V E V E v V E 2 + V E 1 + V E v c 2 ,
d t + d d = 0 .
d t = r s r t = r s r t + ( γ L 1 ) V L V L ( r s r t ) V L 2 γ L V L c ( c t s c t t ) ,
γ L = 1 1 V L 2 c 2 .
r s r t = d t .
d t = d t + ( γ L 1 ) V L V L d t V L 2 γ L V L c d t .
( d t ) 2 = d t d t = d t 2 + 2 ( γ L 1 ) ( V L d t ) 2 V L 2 2 γ L V L d t c d t + ( γ L 2 2 γ L + 1 ) V L 2 ( V L d t ) 2 V L 4 2 ( γ L 2 γ L ) V L 2 V L d t c d t + γ L 2 V L 2 c 2 d t 2 .
γ L 2 1 V L 2 = γ L 2 c 2 , 1 + γ L 2 V L 2 c 2 = γ L 2 ,
d t = γ L d t ( 1 V L d t c d t ) .
n t = d t d t = n t + ( γ L 1 ) V L ( V L n t ) V L 2 γ L V L c γ L ( 1 V L n t c ) ,
V L n t = V L n t V L 2 c 1 V L n t c .
w = w + ( γ L 1 ) V L V L w V L 2 γ L V L γ L ( 1 V L w c 2 ) .
w n t + ( γ L 1 ) V L n t V L w V L 2 γ L V L n t γ L ( 1 V L w c 2 ) = c 2 Δ f f 0 1 + 1 2 Δ f f 0 .
w [ n t + ( γ L 1 ) V L V L n t V L 2 γ L ] = V L n t + ( 1 V L w c 2 ) c 2 Δ f f 0 1 + 1 2 Δ f f 0 ,
w [ n t + ( γ L 1 ) V L V L n t V L 2 γ L + V L c 2 c 2 Δ f f 0 1 + 1 2 Δ f f 0 ] = V L n t + c 2 Δ f f 0 1 + 1 2 Δ f f 0 .
A = n t γ L + ( 1 1 γ L ) V L V L n t V L 2 + V L c 2 c 2 Δ f f 0 1 + 1 2 Δ f f 0 .
w A = V L n t + c 2 Δ f f 0 1 + 1 2 Δ f f 0 .
A = n t γ L 2 ( 1 V L n t c ) V L c + V L c 2 c 2 Δ f f 0 1 + 1 2 Δ f f 0 .
w = w + ( γ E 1 ) V E V E w V E 2 + γ E V E γ E ( 1 + V E w c 2 ) .
w A + ( γ E 1 ) w V E V E A V E 2 + γ E V E A γ E ( 1 + V E w c 2 ) = V L n t + c 2 Δ f f 0 1 + 1 2 Δ f f 0 .
w [ A + ( γ E 1 ) V E V E A V E 2 γ E V E c 2 ( c 2 Δ f f 0 1 + 1 2 Δ f f 0 + V L n t ) ] = c 2 Δ f f 0 1 + 1 2 Δ f f 0 + V L n t V E A .
B = [ A + ( γ E 1 ) V E V E A V E 2 γ E V E c 2 ( c 2 Δ f f 0 1 + 1 2 Δ f f 0 + V L n t ) ] .
w B = c 2 Δ f f 0 1 + 1 2 Δ f f 0 + V L n t V E A .
n t = n t , A = n t , B = A = n t .
w n t = λ 2 Δ f + ( V L V E ) n t .
A n t ( 1 + V L n t c ) V L c ,
B A .
w ( n t V L n t n t V L c ) = c 2 Δ f f 0 1 + 1 2 Δ f f 0 + ( V L V E ) ( n t V L n t n t V L c ) .
w [ n t ( 1 + V L n t c ) V L c + V L V E c 2 c 2 Δ f f 0 1 + 1 2 Δ f f 0 + n t c 2 ( ( ( V L n t ) 2 V L 2 V E 2 2 ) ) + V E c 2 ( V L n t + V E n t 2 ) ] = ( 1 + V E V L c 2 ) c 2 Δ f f 0 1 + 1 2 Δ f f 0 + ( V E V L ) [ n t ( 1 + V L n t c ) V L c + n t c 2 ( ( ( V L n t ) 2 V L 2 ) ) ] .
R e = 6.3780000 × 10 6 m , earth radius ,
h = 4.0000000 × 10 5 m , nominal altitude ,
a = R e + h = 6.778000 × 10 6 m , orbit semimajor axis ,
e = 0.00200 eccentricity ,
ω = 1.600 rad , altitude of perigee ,
Ω = 1.309 rad , angle of ascending line of nodes ,
i = 60.00 ° , orbit inclination ,
t p = 3400.00 s , time of perigee passage .
ϕ ( t ) = 105 π 180 + ω e t .
E e sin E = G M a 3 ( t t p ) ,
cos f = cos E e 1 e cos E .
r = a ( 1 e cos E ) .
x = r ( cos Ω cos ( f + ω ) cos I sin Ω sin ( f + ω ) ) ,
y = r ( sin Ω cos ( f + ω ) + cos I cos Ω sin ( f + ω ) ) ,
z = r sin I sin ( f + ω ) .
r L = { 3490685.517 , 4268379.510 , 3926595.155 } m ,
V L = { 1415.72415 , 5709.87278 , 4928.19131 } m s .
c t s = c t t + 1 c r s ( t s ) r t ( t t ) ,
t s = 13750.0023732965 s .
r E = { 3296533.996 , 3606135.417 , 4099699.375 } m ,
V E = { 262.88727 , 240.31733 , 0.00000 } m s .
d t = { 194154.8809 , 662230.5410 , 173092.5238 } m ,
d t = 711481.8543 m .
w = { 1095.2220 , 5447.7191 , 4897.9803 } m s .
Δ f f = 3.9783903 × 10 5 ,
c 2 Δ f f 0 1 + 1 2 Δ f f 0 = c ( 1.9891555694 × 10 5 ) .
A = { 0.272888029028 , 0.93077643041 , 0.24328452306 } .
B = { 0.272888029028 , 0.93077643041 , 0.24328452306 } .
B = { 0.50449226 E , 0.79099629 N } .
c 2 Δ f f + ( V L V E ) n t = ( 5.963456998 × 10 3 + 5.975359093 × 10 3 ) m s .
Δ f f = 3.70437752 × 10 5 ,
0.5044922625 w e + 0.7909962943 w n = 11.952212693 m s .
B = { 0.4981724053 , 0.33958394090 , 0.797813889 } ,
B = { 0.3933068009 E , 0.8628684348 N } .
0.39330680087 w e 0.86286843481 w n = 60.22015497 m s .
( w e , w n ) = ( 50.000000 , 46.999999 ) m s ,
V L ( t d ) = V L ( t t ) + g ( t d t t ) ,
r L ( t d ) = r L ( t t ) + V L ( t t ) ( t d t t ) + 1 2 g ( t d t t ) 2 ,
t s t t = 1 c r s ( t s ) r t ( t t ) ,
d t s d t t = 1 c ( r s ( t s ) r t ( t t ) ) ( d r s ( t s ) d r t ( t t ) ) r s ( t s ) r t ( t t ) .
n t = r s ( t s ) r t ( t t ) r s ( t s ) r t ( t t ) .
d r s = d r s d t s d t s = w d t s , d r t = V L ( t t ) d t t .
d t s d t t = n t ( w d t s V L ( t t ) d t t ) c ,
d t s ( 1 n t w c ) = d t t ( 1 n t V L ( t t ) c ) .
( c d τ t ) 2 = ( 1 + 2 Φ ( r t ) c 2 ) ( c d t t ) 2 ( d x 2 + d y 2 + d z 2 ) ,
= ( 1 + 2 Φ ( r t ) c 2 V L ( t t ) 2 c 2 ) ( c d t t ) 2 ,
d τ t = 1 + 2 Φ ( r t ) c 2 V L ( t t ) 2 c 2 d t t .
d t t = 1 f 0 1 + 2 Φ ( r t ) c 2 V L ( t t ) 2 c 2 ,
d t s ( 1 n t w c ) = ( 1 n t V L ( t t ) c ) f 0 1 + 2 Φ ( r t ) c 2 V L ( t t ) 2 c 2 .
t d t s = 1 c r d ( t d ) r s ( t s ) ,
d t d d t s = 1 c ( r d ( t d ) r s ( t s ) ) ( d r d ( t d ) d r s ( t s ) ) r d ( t d ) r s ( t s ) .
n d = r d ( t d ) r s ( t s ) r d ( t d ) r s ( t s ) ,
d t s ( 1 n d w c ) = d t d ( 1 n d V L ( t d ) c ) .
d τ d = 1 f = 1 + 2 Φ ( r d ) c 2 V L ( t d ) 2 c 2 d t d ,
d t d = 1 f 1 + 2 Φ ( r d ) c 2 V L ( t d ) 2 c 2 .
d t s ( 1 n d w c ) = ( 1 n d V L ( t d ) c ) f 1 + 2 Φ ( r d ) c 2 V L ( t d ) 2 c 2 .
f f 0 = 1 + Φ ( r t ) c 2 V L ( t t ) 2 c 2 1 + Φ ( r d ) c 2 V L ( t d ) 2 c 2 ( 1 n t w c ) ( 1 n d w c ) ( 1 n d V L ( t d ) c ) ( 1 n t V L ( t t ) c ) .
1 + Φ ( r t ) c 2 V L ( t t ) 2 c 2 1 + Φ ( r d ) c 2 V L ( t d ) 2 c 2 1 + 2 ( t d t t ) c 2 V L g ,
f f 0 = ( 1 n t w c ) ( 1 n d w c ) ( 1 n d V L ( t d ) c ) ( 1 n t V L ( t t ) c ) .
1 + Δ f f 0 = 1 n t w c + n d w c n d V L ( t d ) c + n t V L ( t t ) c .
w n t = λ 2 Δ f + ( V L V E ) n t + 1 2 n t g ( t d t t ) .
1 2 9 × 8.4 × 10 3 m s 1 = 0.04 m s 1 ,
F = f f 0 ( 1 n t V L ( t t ) c ) ( 1 n d V L ( t d ) c ) .
F = ( 1 n t w c ) ( 1 n d w c ) ,
w ( n t F n d ) = c ( 1 F ) .
c ( 1 F ) γ E ( 1 + V E w c 2 ) = w ( n t F n d ) + ( γ E 1 ) w V E V E ( n t F n d ) V E 2 + γ E V E ( n t F n d ) .
w ( ( n t F n d ) γ E c ( 1 F ) V E c 2 + ( γ E 1 ) V E V E 2 V E ( n t F n d ) ) = γ E c ( 1 F ) γ E V E ( n t F n d ) .
c ( 1 f f 0 ( 1 n t V L ( t t ) c + n d V L ( t d ) c ) ) V E ( n t n d ) + n t V L ( t t ) n d V d ( t d ) V E n t + V E n d +
n d g ( t d t t ) c .08 m s 1 ,
2 γ L 2 ( 1 V L n t c ) ( 1 + 1 2 Δ f f 0 ) A = n t F n d ,
2 γ L 2 ( 1 V L n t c ) ( 1 + 1 2 Δ f f 0 ) ( c 2 Δ f f 0 1 + 1 2 Δ f f 0 + V L n t ) = c ( 1 F ) .
V L ( t d ) = V L ( t t ) .
d d + d t = V L ( t d t t ) ,
d d = d t 1 2 V L n t c + V L 2 c 2 1 V L 2 c 2 .
d d = d t + 2 V L d t c 1 V L n t c 1 V L 2 c 2 .
w n t = c 2 Δ f f 0 + V L n t V E n t ,
n d = d d d d n t ( 1 + 2 V L n t c ) + 2 V L c .
( w + V E V L ) ( n t ( 1 + V L n t c ) V L c ) = c 2 Δ f f 0 1 + 1 2 Δ f f 0 .
V L c = 1 V E 2 c 2 V L c + ( 1 1 V E 2 c 2 ) V E V E V L V E 2 c V E c 1 V E V L c 2 .
r = r + ( γ L 1 ) V L V L r V L 2 + γ L V L c c t
c t = γ L ( c t + V L r c ) ,
γ L = 1 1 V L 2 c 2 .
r t = r t + ( γ L 1 ) V L V L r t V L 2 + γ L V L c c t t ,
r s = r s + ( γ L 1 ) V L V L r s V L 2 + γ L V L c c t s .
d t = r s r t = d t + ( γ L 1 ) V L V L d t V L 2 + γ L V L c ( c t s c t t ) .
d t = d t + ( γ L 1 ) V L V L d t V L 2 + γ L V L c d t .
d t = γ L d t ( 1 + V L d t c d t ) ,
d t = d t + ( γ L 1 ) V L V L d t V L 2 γ L V L c d t ,
d t = γ L d t ( 1 V L d t c d t ) .
w d t c d t ,
w = w + ( γ L 1 ) V L V L w V L 2 γ L V L γ L ( 1 V L w c 2 ) .
w d t d t = c 2 Δ f f 0 1 + 1 2 Δ f f 0 = ( w + ( γ L 1 ) V L V L w V L 2 γ L V L ) ( d t + ( γ L 1 ) V L V L d t V L 2 γ L V L c d t ) γ L 2 d t ( 1 V L w c 2 ) ( 1 V L d t c d t ) .
γ L 2 c 2 Δ f f 0 1 + 1 2 Δ f f ( 1 V L w c 2 ) = w d t c d t ( 1 V L d t c d t ) γ L 2 w V L c + γ L 2 V L d t d t + V L 2 c ( 1 V L d t c d t ) .
w n t γ L 2 ( 1 V L n t c ) w V L c 2 ( c c 2 Δ f f 0 1 + 1 2 Δ f f 0 ) = c 2 Δ f f 0 1 + 1 2 Δ f f 0 + V L n t V L 2 c ( 1 V L n t c ) ,
D t = n t γ L 2 ( 1 V L n t c ) V L c 2 ( c c 2 Δ f f 0 1 + 1 2 Δ f f 0 ) .
w D t = c 2 Δ f f 0 1 + 1 2 Δ f f 0 + V L n t V L 2 c ( 1 V L n t c ) .
V E c = 1 V L 2 c 2 V E c + ( 1 1 V L 2 c 2 ) V L V L V E V L 2 c V L c 1 V L V E c 2 .
N ̂ sin ( ψ ) = V L × ( V E ) V L 2 .
N ̂ sin ( ψ ) = V L × ( V E ) 2 c 2 .
w ( n t + V L c ( 1 V L n t c ) c 2 Δ f f 0 1 + 1 2 Δ f f 0 V L c ( 1 V L n t c ) ) = c 2 Δ f f 0 1 + 1 2 Δ f f 0 ( 1 + V L 2 2 c 2 V L n t c ) + ( 1 + V L 2 2 c 2 ) V L n t V L 2 c .
w ( n t V E c ( 1 + V E n t c ) c 2 Δ f f 0 1 + 1 2 Δ f f 0 + V E c ( 1 + V E n t c ) ) = c 2 Δ f f 0 1 + 1 2 Δ f f 0 ( 1 + V E 2 2 c 2 + V E n t c ) ( 1 + V E 2 2 c 2 ) V E n t V E 2 c .
r t = r t .
( x s y s z s ) = ( 1 ω E ( t s t t ) 0 ω E ( t s t t ) 1 0 0 0 1 ) ( x s y s z s ) .
( r t r s ) 2 = ( r t r s ) 2 + 2 ( t s t t ) ω ( r t × r s ) .
A = 1 2 r t × r s ,
( r t r s ) 2 = ( r t r s ) 2 + 4 ( t s t t ) ω A ,
r t r s = r t r s + 2 ( t s t t ) ω A r t r s .
t s t t = 1 c r t r s 1 c r t r s + 2 ω E A c 2 .
r t r s = r t r s + 2 ω E A c .
V L = V L + ω × r t .

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