special relativity corrections for space-based lidars

pocptqtic

Special relativity corrections for space-based lidars

Venkata S. Rao Gudimetla and Michael J. Kavaya

The theory of special relativity is used to analyze some of the physical phenomena associated withspace-based coherent Doppler lidars aimed at Earth and the atmosphere. Two important cases of diffusescattering and retroreflection by lidar targets are treated. For the case of diffuse scattering, we showthat for a coaligned transmitter and receiver on the moving satellite, there is no angle between trans-mitted and returned radiation. However, the ray that enters the receiver does not correspond to aretroreflected ray by the target. For the retroreflection case there is misalignment between the trans-mitted ray and the received ray. In addition, the Doppler shift in the frequency and the amount of tipfor the receiver aperture when needed are calculated. The error in estimating wind because of theDoppler shift in the frequency due to special relativity effects is examined. The results are then appliedto a proposed space-based pulsed coherent Doppler lidar at NASA’s Marshall Space Flight Center forwind and aerosol backscatter measurements. The lidar uses an orbiting spacecraft with a pulsed lasersource and measures the Doppler shift between the transmitted and the received frequencies to deter-mine the atmospheric wind velocities. We show that the special relativity effects are small for theproposed system. © 1999 Optical Society of America

OCIS codes: 010.0010, 010.3640, 010.7030, 280.0280, 350.5270, 350.6090.

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1. Introduction

The need to accurately profile tropospheric vectorwinds versus height remotely from space is growingin importance with agencies such as NASA, the Na-tional Oceanic and Atmospheric Administration~NOAA!, the U.S. Department of Defense, the Euro-

ean Space Agency ~ESA!, and the U.S. Departmentf Energy. NASA’s, NOAA’s, and ESA’s interest in-ludes research on global climate change and im-roved weather prediction.1–4 A most promisingechnique for taking the measurement, given re-uired wind measurement specifications, the naturalropospheric environment in which the remote sens-ng must be accomplished, and the limited resourcesonnected with space flight, is coherent ~heterodyne!

detection laser radar2 ~lidar!.One can obtain vector winds by probing each air-

When this research was performed, V. S. R. Gudimetla was withthe Department of Electrical and Computer Engineering, OregonGraduate Institute, P.O. Box 91000, Portland, Oregon 97291-1000.He is now with Ansoft Corporation, 669 River Drive, Suite 200,Elmwood Park, New Jersey 07407-1361. His e-mail address [email protected]. M. J. Kavaya is with the Global Hydrol-ogy and Climate Center, NASA Marshall Space Flight Center,Mail Stop SD60, Huntsville, Alabama 35812.

Received 11 June 1999; revised manuscript received 11 June1999.

0003-6935y99y306374-09$15.00y0© 1999 Optical Society of America

6374 APPLIED OPTICS y Vol. 38, No. 30 y 20 October 1999

mass from two or more perspectives. Each individ-ual line-of-sight measurement consists oftransmitting a laser pulse from the lidar instrumenttoward the troposphere, collecting photons that scat-ter off natural aerosol particles in the air and returnto the lidar, determining the Doppler shift of thebackscattered photons after removing the nonwindDoppler shifts that are due to lidar and Earth mo-tions, and by use of the geometry and the echo ordelay time of the photons to assign measurementlocation and height. Backscattered photons fromthe ground and oceans may be utilized either to ob-tain or to confirm nonwind motion removal and mea-surement altitude above ground level assignments.The coherent lidar signal-to-noise ratio ~SNR!, andhence the performance, drops quickly with the in-crease in misalignment angle between the directionof the transmitted photons and the direction of thereceiver at the time the backscattered photons reen-ter the lidar.5,6 For example, a coherent lidar with2.06-mm wavelength photons and a 0.22-m receiveraperture will lose on average 3 dB of SNR for a bore-sight misalignment angle of 7.1 mrad ~1.5 arc sec!.

herefore good alignment is important.Coherent lidar profiling of wind velocity is an es-

ablished successful technique from ground, ship,nd airborne platforms.7–13 The NASA Space

Readiness Coherent Lidar Experiment ~SPARCLE!mission will be the first space-based measurement,14

which involves two simultaneous new parameter re-

Wr

atsesscrt

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f

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atagn

gimes: ~1! much higher lidar to aerosol target rela-tive motion ~e.g., 7733-mys tangential velocity for a300-km orbit! and ~2! much longer signal echo timesthat are due to much longer sensing ranges ~e.g., aslant range to the ground as great as 344 km andhence a 2.3-ms echo time for a 300-km orbit, and a30-deg laser beam nadir angle at the lidar!. Thesenew parameter regimes led us to consider specialrelativity effects and the following questions:

~1! Will any photons scattered from the target ~e.g.,aerosol particles, cloud particles, land, ocean! bothreenter the orbiting lidar receiver and be in align-ment with the coaligned transmitteryreceiver axis?

~2! If the answer to ~1! is no, we ask the following:ill the alignment condition be achieved with a cor-

ection to the receiver optics design ~e.g., tilt!?~3! How does special relativity change the incident

nd the scattering angles of the photons with respecto the scattering target? Is the term backscattertill appropriate? Are the angle differences smallnough that published estimates of target and back-catter reflectivity are still valid? ~The term back-catter is not used uniformly in the literature. Itould mean the exact retroreflected ray, or it mightefer to all rays in the general retroreflected direc-ion. We use the latter definition here.!

~4! Is the wind velocity measurement affected?an relativistic effects be accounted for and removed?~5! Can retroreflectors on the ground be considered

or instrument validationycalibration?

In this paper the above questions are addressed inonditions of a flat Earth and special relativity. Wereat the translation effects of the moving spacecraftyidar and not the circular orbit effects ~nadir directionipping! or the effects of a continuously moving lidarcanner. Although in the literature one finds sev-ral papers on the scattering off a movingoundary,15–35 the emphasis in those papers had been

on the reflections off mirrors and dielectric bound-aries but not on the backscattered and the diffuselyscattered rays that are more relevant to remote-sensing systems. Only Young36 considered such aproblem, and in his results he discussed only theDoppler shift in frequency and not the other issues inthis paper. Hence the problem of laser scattering offa target when the source and receiver are moving wasanalyzed in the following, with relativistic electrody-namics, and applied to the case of space-based coher-ent lidar. We are interested in developing analyticalexpressions for the locations, angles, and Dopplershifts of the laser light scattering at the target and ofthe received light at the lidar. The two importantcases of diffuse scattering and retroreflection by thetarget are treated because they are found to be usefulfor space applications.

The organization of the paper is as follows: InSubsection 1.A we state all the basic assumptionsalong with the Lorentz transformations that areneeded to introduce special relativistic effects in theanalysis and we also describe the path geometry. In

Subsection 1.B first we calculate the incident angleand point of contact of a ray transmitted from thesatellite in the fixed coordinate system of the ground.In Subsection 1.C we analyze the case of diffuse scat-tering by the target and show that the angles of re-flection at the target differ from the angles ofincidence. In Subsection 1.D we examine especiallythe ray that enters the untipped receiver and, inSubsection 1.E, the case of retroreflection, which isidentical to one of the many rays in the case of diffusescattering. In Section 2 we examine the Dopplershift between the transmitted and the received fre-quencies. We also examine the relativistic effects onwind measurements. Our conclusions are given inSection 3.

A. Assumptions, Formulation, and Path Geometry

Initially we assume that the scattering ground is flatand that a satellite carrying a Doppler lidar windsounder moves with a horizontal velocity V ~along the

axis parallel to the ground! at a height H above theround ~Fig. 1!. For the proposed wind sounder, thePARCLE Space Shuttle Mission,14 the Doppler lidar

sends light rays ~2.051-mm wavelength! with a nadirngle of 30° and an azimuth angle of 45° toward thetmosphereyground and collects the return radiationor the measurement of winds. SPARCLE is capa-le of any azimuth angle. In this paper an azimuthngle of 45° is used. The proposed satellite velocitys 7733 mys at a height of 300 km above the ground.or SPARCLE most of the return radiation comes

rom diffuse scattering by atmospheric aerosols, theround, or the sea surface. Other possibilities arehe use of retroreflectors as in the case of Geodynamicaser Ranging Systems. However, most practicalituations correspond to return by diffuse scatterersnd so this case is considered first.Results for the case of retroreflection are developed

s a special case for this analysis. As a first step inhe analysis we calculate the path of a light ray from

moving source ~satellite lidar! to the stationaryround, as observed from the ground ~whose coordi-ate system does not move!. In subsequent sections

Fig. 1. Lidar and target geometry and coordinate systems.

20 October 1999 y Vol. 38, No. 30 y APPLIED OPTICS 6375

tK

0

s

t

ci

y

Ttn

6

the paths of the diffusely scattered or retroreflectedrays from the ground toward the satellite are ana-lyzed.

Figure 1 shows two coordinate systems. The KGsystem is a fixed coordinate system with coordinatesgiven by xG, yG, zG, and tG. The KV system describeshe satellite lidar motion and moves with respect toG at velocity V along the xV axis. A laser transmit-

ter is located at the origin of the KV system. Thecoordinates in KV are given as xV, yV, zV, and tV,respectively. Assume at tG 5 tV 5 0 that point ~xV 5, yV 5 0, zV 5 0! of the KV coordinate system passes

the point ~xG 5 0, yG 5 0, zG 5 2H! of coordinateystem KG. This assumption is needed for the va-

lidity of the following Lorentz transformations, whichrelate the coordinates of these two systems as15 ~c ishe light velocity, a 5 Vyc!

xG 5xV 1 VtV

~1 2 a2!1y2 , yG 5 yV,

zG 5 zV 2 H, ctG 5ctV 1 axV

~1 2 a2!1y2 . (1)

Inversion formulas are given by

xV 5xG 2 VtG

~1 2 a2!1y2 , yV 5 yG,

zV 5 zG 1 H, ctV 5ctG 2 axG

~1 2 a2!1y2 . (2)

Use of direction cosines facilitates the analysis inthree dimensions, and the direction cosines are de-fined as

cos~cx! 5 sin~u!cos~f!,

cos~cy! 5 sin~u!sin~f!,

cos~cz! 5 cos~u!. (3)

Conversely, the azimuth and the elevation angles aregiven by

tan~f! 5cos~cy!

cos~cx!, cos~u! 5 cos~cz!. (4)

Equations ~3! and ~4! help the transformation for thedirection cosines to the spherical coordinate angles~elevationynadir and azimuth angles! and vice versa.

B. Calculation of the Coordinates and Angles where theLight Transmitted from a Moving Satellite Hits the Ground

Although the proposed problem is equivalent to afixed sourceyreceiver ~lidar! with a moving target, thefollowing analysis is done with a moving sourceyre-ceiver and fixed target. Consider a light signaltransmitted from the KV system in a direction withspherical coordinate angles uVt and fVt ~Fig. 1! to-ward the fixed system on the ground. The nadir uVtis the angle between the zV axis and the transmittedray, and the azimuth fVt is the angle between the xVaxis and the projection of the ray on the x–y plane.


Let the direction cosines of the transmitted ray in theKV system be called cVtx, cVty, and cVtz, where sub-script t indicates the transmitter coordinate systemand the subscript x, y, or z refers to the axis in thatcoordinate system.

After the light ray is emitted at tV 5 0, coordinatesof its trajectory in the KV system are given as

xV 5 ctV cos~cVtx! 5 ctV sin~uVt!cos~fVt!,

yV 5 ctV cos~cVty! 5 ctV sin~uVt!sin~fVt!,

zV 5 ctV cos~cVtz! 5 ctV cos~uVt!. (5)

One can calculate the time coordinate of the point ofcontact of the light ray with the ground by solving fortV 5 tVg when zV 5 H. In this case

tVg 5H

c cos~cVtz!5

Hc cos~uVt!

, (6)

and when this result is used in Eq. ~5!, the spatialoordinates of the point of incidence on the ground ~Gn Fig. 1! in KV are calculated as

xVg 5 Hcos~cVtx!

cos~cVtz!5 H tan~uVt!cos~fVt!,

yVg 5 Hcos~cVty!

cos~cVtz!5 H tan~uVt!sin~fVt!,

zVg 5 H. (7)

We obtained the corresponding coordinates in the KGsystem for the point of incidence of the light ray onthe ground by using Eqs. ~7! and ~2!:

xGg 5 Ha 1 sin~uVt!cos~fVt!

~1 2 a2!1y2 cos~uVt!5 H

a 1 cos~cVtx!

~1 2 a2!1y2 cos~cVtz!,

Gg 5 H tan~uVt!sin~fVt! 5 Hcos~cVty!

cos~cVtz!, zGg 5 0,

(8)

tGg 5Hc

1 1 a sin~uVt!cos~fVt!

~1 2 a2!1y2 cos~uVt!

5Hc

1 1 a cos~cVtx!

~1 2 a2!1y2 cos~cVtz!. (9)

he length of the light ray r between the origin ofransmission and point G is given in a fixed coordi-ate system as

rG 5 c~tGg 2 0! 5H

~1 2 a2!1y2

@1 1 a cos~cVtx!#

cos~cVtz!

5H

~1 2 a2!1y2

@1 1 a sin~uVt!cos~fVt!#

cos~uVt!. (10)

g

cau

a

t

U

b

Uti

Tt

The direction cosines of the received light ray at theground, denoted as cGgx, cGgy, and cGgz, in the fixedcoordinate system KG are given by

cos~cGgx! 5xGg 2 0

r5

a 1 cos~cVtx!

1 1 a cos~cVtx!

5a 1 sin~uVt!cos~fVt!

1 1 a sin~uVt!cos~fVt!,

cos~cGgy! 5yGg 2 0

r5

~1 2 a2!1y2 cos~cVty!

1 1 a cos~cVtx!

5~1 2 a2!1y2 sin~uVt!sin~fVt!


cos~cGgz! 5zGg 2 ~2H!

r5

~1 2 a2!1y2 cos~cVtz!

1 1 a cos~cVtx!

5~1 2 a2!1y2 cos~uVt!

1 1 a sin~uVt!cos~fVt!. (11)

By inverting Eq. ~11!, one can also calculate the di-rection cosines in the moving coordinate system if thecorresponding values in the fixed coordinate systemare known. From the above results we get coordi-nate angles uGg and fGg for the ray received at theround as

cos~uGg! 5 cos~cGgz! 5~1 2 a2!1y2 cos~uVt!


tan~fGg! 5cos~cGgy!

cos~cGgx!5

~1 2 a2!1y2 sin~uVt!sin~fVt!

a 1 sin~uVt!cos~fVt!.

(12)

When the data for the proposed wind sensor~SPARCLE! ~uVt 5 30° and fVt 5 45°! are used, theorresponding spherical coordinate angles for the rayt the ground in the ground coordinate system areGg 5 30.0009° and fGg 5 44.9979°. Compared

with a nonmoving lidar, V 5 0, the incident nadirngle on the target differs by 0.0009° 5 16 mrad 5 3.2

arc sec, and the incident azimuth angle differs by0.0021° 5 37 mrad 5 7.6 arc sec.

C. General Location of Return and Angle of Arrival at theSatelliteyLidar for a Given Ray after Diffuse Scattering

In general, after diffuse scattering there is an ensem-ble of rays in different directions. Consider a rayhaving directional cosines cGdx, cGdy, and cGdzamong this ensemble ~Fig. 2!, where d of the sub-script indicates diffuse reflection and the third letterindicates the coordinate axis. The coordinates ofthis ray in the fixed coordinate system at time tG aregiven as

xG 5 c~tG 2 tGg!cos~cGdx! 1 xGg,

yG 5 c~tG 2 tGg!cos~cGd y! 1 yGg,

zG 5 c~tG 2 tGg!cos~cGdz!. (13)

At the point of return, zG 5 2H in the KG system andthe corresponding time coordinate, tG 5 tGret, is ob-ained from Eqs. ~13! and ~9! as

tGret 5 tGg 2H

c cos~cGdz!5 H

1 1 a cos~cVtx!

c~1 2 a2!1y2 cos~cVtz!

2H

c cos~cGdz!. (14)

sing this result in Eq. ~13! along with Eqs. ~8! and~9! for xGg, yGg, and tGg gives the coordinates of thepoint of return ~subscript ret implies that the valueselong to the return radiation! as

xGret 5 2Hcos~cGdx!

cos~cGdz!1 H

a 1 cos~cVtx!

~1 2 a2!1y2 cos~cVtz!, (15)

yGret 5 2Hcos~cGd y!

cos~cGdz!1 H

cos~cVty!

cos~cVtz!,

zGret 5 2H. (16)

sing the Lorentz transformations, one can calculatehe corresponding coordinates of the point of returnn moving system KV as

xVret 5 Ha 2 cos~cGdx!

~1 2 a2!1y2 cos~cGdz!1 H

cos~cVtx!

cos~cVtz!,

yVret 5 yGret 5 2Hcos~cGd y!

cos~cGdz!1 H

cos~cVty!

cos~cVtz!.

zVret 5 0. (17)

he length of the returned diffuse ray between thesewo points can be calculated as

rV 5 @uxVret 2 xVgu2 1 uyVret 2 yVgu2 1 uzVret 2 zVgu2#1y2

5 H21 1 a sin~uGd!cos~fGd!

~1 2 a2!1y2 cos~uGd!

5 H21 1 a cos~cGdx!

~1 2 a2!1y2 cos~cGdz!. (18)

Fig. 2. Path geometry for the diffusely reflected ray.


a

wttmtg

Tt

r2naio

mtS

6

Using the above results we calculate the directioncosines of the return radiation as

cos~cVretx! 5xVret 2 xVg

rV5

a 2 cos~cGdx!

21 1 a cos~cGdx!,

cos~cVrety! 5yVret 2 yVg

rV5 2

~1 2 a2!1y2 cos~cGd y!

21 1 a cos~cGdx!,

cos~cVretz! 5zVret 2 zVg

rV5 2

~1 2 a2!1y2 cos~cGdz!

21 1 a cos~cGdx!. (19)

From the above directional cosines, the receivingspherical angles are calculated as

cos~uVret! 5 cos~cVretz!

5~1 2 a2!1y2 cos~uGd!

1 2 a sin~uGd!cos~fGd!,

tan~fVret! 5cos~cVrety!

cos~cVretx!

5 2~1 2 a2!1y2 sin~uGd!sin~fGd!

a 2 sin~uGd!cos~fGd!. (20)

The above results indicate that the angles of inci-dence of the return radiation as measured in themoving coordinate system differ from the angles ofreflection as measured in the fixed coordinate system.

However, we are interested in examining two par-ticular rays; the first is the ray among all the diffuselyscattered rays at the target that enters the lidar re-ceiver at its new location, and the second is the raythat is retroreflected back along the line of incidenceat the target. These two issues are addressed inSubsections 1.D and 1.E.

D. Direction of the Scattered Ray that Enters the MovingLidar

Among all the rays of diffusely scattered radiation,the only important ray is the ray that does not missthe lidar receiver. To calculate the direction of thisray, we start with Eq. ~17! and require that

xVret 5 0, yVret 5 0, zVret 5 0. (21)

Solving Eqs. ~21! ~and also using the fact that the sumof the squares of all three direction cosines is unity!gives the following solution for the direction of the rayin the KG system as ~this important solution is indi-cated by subscripts d1x, d1y, and d1z!

cos~cGd1x! 5a 2 cos~cVtx!

1 2 a cos~cVtx!,

cos~cGd1y! 5 2~1 2 a2!1y2 cos~cVty!

1 2 a cos~cVtx!,

cos~cGd1z! 5 2~1 2 a2!1y2 cos~cVtz!

1 2 a cos~cVtx!. (22a)

The corresponding direction cosines of this radiationat the receiver in the moving KV coordinate system


were calculated by use of Eq. ~19!, and these cosinesre given by

cVretx 5 cVtx 2 p, cVrety 5 cVty 2 p, cVretz 5 p 2 cVtz,

(23)

hich shows that there is no misalignment betweenhe transmitted ray and the scattered ray that reen-ers the moving lidar. Therefore no SNR loss fromisalignment occurs. Converting the above direc-

ion cosine results into spherical coordinate anglesives ~denoted as uGd1 and fGd1!

cos~uGd1! 5 2cos~uVt!~1 2 a2!1y2


tan~fGd1! 5 2sin~uVt!sin~fVt!~1 2 a2!1y2

a 2 sin~uVt!cos~fVt!. (22b)

An alternative approach is to start with the factthat, when the receiver aperture is not tipped, it seesa returning ray whose direction cosines ~cVretx, cVrety,and cVretz! satisfy the relations cVtx 5 cVretx 1 p,cVty 5 cVrety 1 p, and cVtz 5 p 2 cVretz as in Eq. ~23!.

hen calculate the corresponding direction cosines ofhe ray at the scatterer in the fixed KG coordinate

system, which leads to the same results as in Eqs.~22a!. When we convert these direction cosines intothe KV coordinate system by using Eq. ~11!, the cor-esponding direction cosines are calculated to becos~cVtx!, 2cos~cVty!, and 2cos~cVtz!. Since theegative sign in front of the direction cosines can beccounted for by reversal of the direction of the ray, its concluded that there is no misalignment with theriginal ray.Thus, in the case of a diffuse target, for any trans-ission direction, scattered light exists that reenters

he moving lidar in perfect alignment. ForPARCLE, when uVt 5 30° and fVt 5 45°, the corre-

sponding angles of the ray that enters the receiverhave coordinate angles of uGd1 5 150.0009° andfGd1 5 225.0021° in the fixed coordinate system.Adjusting for the direction reversal after scattering~180°! and comparing the corresponding angles of theray with uGg and fGg from Subsection 1.B, it is clearthat this is not a retroreflected ray back along the lineof incidence at the target. The scattering angles atthe target that reenter the lidar differ from the ret-roreflection angles by 0.0018° 5 32 mrad 5 6.5 arc secin nadir and by 0.0042° 5 73 mrad 5 15 arc sec inazimuth. Special relativity causes the light to strikethe aerosol particles from a slightly different direc-tion than when there is no motion and then causesthe scattered direction that reenters the lidar to beslightly different from the retroreflection direction.

E. Location of Return and Angle of Arrival forRetroreflected Ray

Next we examine the case of the ray that is retrore-flected along the line of incidence at the target. Thiscorresponds to the cases where space-based lidar sys-tems use retroreflectors on the ground and where

y

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diffusely scattering targets have enhanced reflectivi-ties in the retroreflection direction.

Assume now that at point G on the ground ~xGg,Gg, and zGg! the incident ray is retroreflected back

along the direction of incidence in KG, the fixed coor-inate system ~Fig. 3!. Hence the direction cosines

of this particular backscattered ray are given bycGdx 5 cGgx 1 p, cGdy 5 cGgy 1 p, and cGdz 5 p 2cGgz, where cGgx, cGgy, and cGgz are given by Eqs.~11!. Using these expressions in Eqs. ~14!–~18! givesthe coordinates of the point of return of a retrore-flected ray ~xVret, yVret, and zVret!, the time point ofreturn ~tVret!, and also the length of the light raybetween the point of scattering at the ground and thepoint of return at the satellite ~rV!, all in the movingcoordinate system KV. The coordinates of the pointof return are given as

xVret 5 22Ha1 1 a sin~uVt!cos~fVt!

~1 2 a2!cos~uVt!,

yVret 5 0, zVret 5 0. (24)

Equation ~24! can be expanded as ~assuming a ,, 1!

xVret 5 22Ha

cos~uVt!@1 1 a sin~uVt!cos~fVt! 1 . . .#, (25)

here the first term is due to the translation of theatellite ~the linear effect! and all other terms are dueo the special relativistic correction.

The corresponding direction cosines and sphericaloordinate angles of the return radiation can be cal-ulated by using Eq. ~22!. The spherical coordinatengles are given as

cos~uVret! 5 2~1 2 a2!cos~uVt!

1 1 a2 1 2a sin~uVt!cos~fVt!,

tan~fVret! 5~1 2 a2!sin~uVt!sin~fVt!

2a 1 ~1 1 a2!sin~uVt!cos~fVt!. (26)

Fig. 3. Path geometry for the retroreflected ray.

rom Eqs. ~26!,

cos~uVret! 5 cos~uVt!$1 2 2a sin~uVt!cos~fVt!

2 2a@1 2 2a sin~uVt!cos~fVt!# 1 . . . %, (27)

and no such simple expansion as a power series of ais possible for fVret.

For the proposed system the radiation returned byretroreflection misses the receiver on the satellitelidar by ;17 m ~xVret 5 217.9 m! and has uVret 5149.9982° and FVret 5 224.9958°. Adjusting for thereversal of direction of propagation, these measure-ments correspond to a tilt of 0.0018° 5 31.60 mrad 5.5 arc sec in nadir and a tilt of 0.0042° 5 73.00rad 5 15 arc sec in azimuth. These examples show

hat tipping the receiver aperture is necessary toave perfect alignment of the radiation retroreflectedlong the direction of incidence. Despite the 18-mosition offset, diffractive spreading of the scatteredight may cause a portion of the scattered energy toe collected by the lidar aperture.

2. Doppler and Wave-Number Shifts

Although the above formulation yielded the resultsfor the coordinates of the returning point and anglesof return, it cannot give us expressions relating thefrequency and the wave vectors of transmission tothose of the return in the satellite coordinate system~Doppler shift!. To achieve these results, the waveformulation was used. Also such a formulation pro-vided an alternative proof for the receiving angle re-sults that were calculated above.

A. Formulation

Consider a plane wave moving along the direction ~bxV,byV, bzV! with a radian frequency of vVt in the movingcoordinate system KV. b’s are the components of thewave vector k. The time variable is denoted as tV.

or a receiver on the ground, let this wave have airection ~bxG, byG, bzG! and a frequency vg in the fixed

system KG where the wave numbers satisfy

bxG2 1 byG

2 1 bzG2 5 b2, (28)

The time variable is denoted as tG. Since light ve-ocity is independent of motion of the source, by rea-on of invariance of the phase of the wave,37

vVttV 2 bxVxV 2 byVyV 2 bzVzV

5 vg tG 2 bxGxG 2 byGyG 2 bzGzG. (29)

sing the Lorentz transformations of Eq. ~2! in Eq.29! for xV, yV, zV, and tV, we manipulated the left-

hand side of Eq. ~29! as follows:

vVttV 2 bxVxV 2 byVyV 2 bzVzV

5vVt

~1 2 a2!1y2 StG 2 axG

c D2

bxV

~1 2 a2!1y2 ~xG 2 vtG! 2 byVyG 2 bzVzG


bm

mttm

cb

tacswidca

ar

fm

.

w

ow

6

5 t1

~1 2 a2!1y2 ~vVt 1 bxVV!

2xG

~1 2 a2!1y2 ~bxV 1 abt0! 2 yGbyV 2 zGbzV. (30)

In Eq. ~30! bt0 5 ~vVt!yc. Matching the coefficientsof xG, yG, zG, and tG from the right- and left-handsides of Eq. ~29! yields the following relationshipsetween the wave numbers and the frequencies of theoving and fixed coordinate systems:

vg 5~vVt 1 bxVV!

~1 2 a2!1y2 , bxG 5~bxV 1 abt0!

~1 2 a2!1y2 ,

byG 5 byV, bzG 5 bzV. (31)

Similarly the inversion formulas can be written as

vVt 5~vg 2 bxGV!

~1 2 a2!1y2 , bxV 5~bxG 2 abg0!

~1 2 a2!1y2 ,

bV1 5 byG, bzV 5 bzG, (32)

where bg0 5 vgyc.Using the above direct and inversion formulas, we

analyzed the problem where the transmitter and thereceiver are located on a moving spacecraft and theradiation is returned by the stationary ground ~or at-

osphere! through diffuse scattering or retroreflec-ion. Assume, as above, that a plane wave is beingransmitted at the spherical coordinate angles in theoving coordinate system uVt and fVt as shown in Fig.

1. Let the direction cosine angles be given by cVtx,Vty, and cVtz. Then the corresponding wave num-ers are

bxV 5 bt0 cos~cVtx! 5 bt0 sin~uVt!cos~fVt!,

byV 5 bt0 cos~cVty! 5 bt0 sin~uVt!sin~fVt!,

bzV 5 bt0 cos~cVtz! 5 bt0 cos~uVt!. (33)

Using Eq. ~31!, we can calculate the received fre-quency at the ground vg as

vg 5~vVt 1 bxVV!

~1 2 a2!1y2

5 vVt

1 1 a sin~uVt!cos~fVt!

~1 2 a2!1y2

5 vVt

1 1 acos~cVtx!

~1 2 a2!1y2. (34)

Using Eq. ~31!, one can calculate the wave numbers ofhe incident ray in the fixed coordinate system first,nd from these values the corresponding sphericaloordinate angles can be calculated. Although nothown here for reasons of brevity, these calculationsere carried out, and the final results for the spher-

cal coordinate angles are identical to the expressionserived in the Subsections above. We confined ouralculations in this section to the Doppler shift only,nd the case of diffuse scattering by the ground is


nalyzed first and the case of retroreflection is de-ived as a special case.

B. Diffuse Scattering

For diffuse scattering, consider a plane wave travel-ing with spherical coordinate angles uGd and fGd asshown in Fig. 2. Then its wave numbers in the fixedcoordinate system, bGdx and bGdz, are given by

bGdx 5 bg0 sin~uGd!cos~fGd!,

bGd y 5 bg0 sin~uGd!sin~fGd!,

bGdz 5 bg0 cos~uGd!, (35)

where bg0 5 vgyc and vg is given in Eq. ~34!. Therequency of the diffuse radiation as observed in theoving KV coordinate system, vVret, is given by @Eqs.

~32!#

vVret 5 g~vg 2 bGdxV!.

Using Eqs. ~35! and ~32!, we get

vVret

5 vVt

~1 1 a sinuVt cos fVt!~1 2 a sin uGd cos fGd!

1 2 a2

(36)

Using uGd and the fGd given in Eqs. ~22b! in Eq. ~36!,e can show that

vVret 5 vVt

1 1 a sin uVt cos fVt

1 2 a sin uVt cos fVt. (37)

Equation ~37! relates the received frequency to thetransmitted frequency for the case of diffuse scatter-ing. The difference between the two frequenciesgives the Doppler shift. For the proposed systemwith a wavelength of 2.051 mm, the total Dopplershift is 2.666 GHz, of which 24.32 kHz is the specialrelativistic contribution and the remaining is the lin-ear Doppler shift that is due to satellite motion. The24.32 kHz corresponds to a potential line-of-slight~line-of-sight! velocity error, if not accounted for, ofnly 0.02 mys. This error is very small comparedith desired wind accuracies of 1 mys.

C. Retroreflection

We treat this as a special case of diffuse reflectionwhere the spherical coordinate angles of the back-scattered radiation are given by Eq. ~26!. Usingthese values for uGd and fGd in Eq. ~36!, the fre-quency of the retroreflected wave, as observed by amoving receiver in the KV system, is calculated as

vVret 5 vVt

1 1 a2 1 2a sin uVt cos fVt

1 2 a2

5 wt@1 1 2a sin~uVt!cos~fVt! 1 2a2

1 2a3 sin~uVt!cos~fVt! 1 . . .#. (38)

The second term is the linear Doppler shift, and allthe other terms except the first are special relativistic

udsTSaSpfTi

4. J. M. Vaughan, K. O. Steinwall, C. Werner, and P. H. Flamant,

1

1

1

1

2

2

corrections. For the proposed wind sensor the totalDoppler shift is 2.6679 GHz and the relativistic con-tribution to the Doppler shift in frequency is 194.38kHz, which corresponds to 0.2-mys wind.

3. Summary

In this paper all the equations were derived that areneeded to study the effects of special relativity for aspace-based lidar system where source and receiverare located on a moving spacecraft. In such sys-tems a laser beam is transmitted toward the groundfrom an orbiting spacecraft and is then scattered bya target. Diffusely scattered rays will be collectedby a receiver, adjacent and coaligned with thetransmitter. The derived general results includethe location of the point of return of light with re-spect to the receiver on the moving satellite lidar,spherical coordinate angles of the returning radia-tion, target scattering angles, and the Doppler shiftin frequency. In the case of diffuse scattering bythe atmosphere or the Earth’s surface, it has beenshown that there is no misalignment between acoaligned transmitteryreceiver and the scatteredray that reenters the untipped lidar receiver. Ourcalculations show that a retroreflected ray missesthe receiver on the satellite by a few meters, and thereceiver must be tipped by a few microradians toalign perfectly with the returning radiation. Thespecial relativity correction to the Doppler shift wasfound to be very small. The retroreflection resultsshould be useful for instrument validation and cal-ibration purposes.

This research was performed as part of the 1994NASA–ASEE ~American Society for Engineering Ed-

cation! Summer Faculty Program. V. S. R. Gu-imetla thanks NASA and ASEE for financialupport through the summer faculty fellowship.he authors thank their colleagues at the Marshallpace Flight Center for several useful discussionsnd their encouragement and the NASA Marshallpace Flight Center and the ASEE for generous sup-ort. They also thank Peter Winzer of the Institutur Nachrichtentechnik und Hochfrequenztechnik,echnische Universitat Wien, Austria, for his keen

nterest and a critical reading of the manuscript.

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