Relativistic effects in earth-orbiting Doppler lidar return signals

Transcription

1 3530 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby Relativisti effets in earth-orbiting Doppler lidar return signals Neil Ashby 1,, * 1 Department of Physis, University of Colorado, Boulder, Colorado , USA Present address: Time and Frequeny Division, Mail Stop 847, National Institute of Standards and Tehnology, Boulder, Colorado 8030, USA *Corresponding author: ashby@boulder.nist.gov Reeived February, 007; revised June 7, 007; aepted July 11, 007; posted August 0, 007 (Do. ID 79581); published Otober 18, 007 Frequeny shifts of side-ranging lidar signals are alulated to high order in the small quantities v/, where v is the veloity of a spaeraft arrying a lidar laser or of an aerosol partile that satters the radiation bak into a detetor ( is the speed of light). Frequeny shift measurements determine horizontal omponents of ground veloity of the sattering partile, but measured frational frequeny shifts are large beause of the large veloities of the spaeraft and of the rotating earth. Subtrations of large terms ause a loss of signifiant digits and magnify the effet of relativisti orretions in determination of wind veloity. Spaeraft aeleration is also onsidered. Calulations are performed in an earth-entered inertial frame, and appropriate transformations are applied giving the veloities of satterers relative to the ground. 007 Optial Soiety of Ameria OCIS odes: , , INTRODUCTION Side-ranging laser imaging detetion and ranging (lidar) measurements of wind from spae will probably use a laser transmitter in a spaeraft to illuminate aerosol partiles or moleules that satter the radiation almost exatly bak toward the transmitter. The measured frequeny shift an be proessed to yield the veloity of the sattering partiles relative to the ground. This paper disusses relativisti orretions to the determination of the wind veloity, arising from large spaeraft veloities, earth rotation, and aeleration of the spaeraft toward the earth. Three different referene frames need to be disussed in this onnetion: these are the instantaneous rest frame of the lidar apparatus, alled the lidar rest frame (LRF); the earth-entered loally inertial frame, alled the earth-entered inertial (ECI) frame; and a referene frame at rest relative to the surfae of the earth at the point where the wind is measured, alled the earth referene frame (ERF). We assume that the position, veloity, and aeleration of the spaeraft are known in the ECI frame beause it seems most likely that the spaeraft would be traked, and its orbit determined, in suh a frame. To simplify the notation, we shall denote quantities measured in the ECI frame without primes. We shall regard the ECI frame, a loally inertial frame whose origin falls along with the earth s enter, as the fundamental frame of referene. Figure 1 illustrates the ECI oordinates that have spatial axes in fixed diretions in spae and origin at earth s enter. Figure illustrates the LRF, with origin at the lidar transmitter and axes fixed in spae when observed from the ECI frame. Quantities measured in the LRF will be doubly primed. Sine the objet of the measurement is to determine the wind veloity relative to the ground, a referene frame with origin at the ground point below the satterer is needed; this point is moving with veloity relative to the ECI frame; its axes do not rotate. This frame is alled the ERF; quantities measured with respet to the ERF will be denoted with primes. Thus the wind veloity will be denoted by w, w, or w in the LRF, ERF, or ECI frame, respetively. A diagram of the ERF is shown in Fig. 3. Table 1 summarizes the differenes among the various referene systems. We shall disuss transformations that enable one to determine relations among the radiation paths in the various referene frames. The LRF is the frame in whih the measurement is made, while the ERF is of interest sine the wind veloity in the ERF is the desired goal of the measurement. We assume that in the ECI frame, light travels with speed in all diretions; gravitational slowing of the signal pulses amounts to no more than a few pioseonds and is negleted. No atmospheri refration effets are inluded; the derivations given here fous purely on relativisti effets. The outline of the paper is as follows. In Setion we derive the frational frequeny shift observed in the LRF, assuming the veloity of the apparatus is onstant during the propagation time of the radiation. First-order relativisti orretions are disussed. Setion 3 develops Lorentz transformations in vetor form, needed in Setion 4 to derive higher-order relativisti orretions. We use Lorentz transformations in vetor form as a ompat way of expressing relativisti transformations among the various referene frames [1 3]. This is useful when the relative veloities of the referene frames are in arbitrary diretions. The vetor notation is explained in Setion 3. Nu /07/ /$ Optial Soiety of Ameria

2 Neil Ashby Vol. 4, No. 11/ November 007/J. Opt. So. Am. A 3531 Fig. 1. Earth-entered inertial oordinates. The origin is at earth s enter, and the xyz axes have fixed diretions in spae. For example, the x axis an point toward the vernal equinox with the z axis parallel to earth s angular veloity vetor. The positions of the sattering partile and the spaeraft are respetively denoted by r and r L in these oordinates. Fig.. Lidar rest frame. This is defined with origin at the lidar transmitter. Quantities measured in this frame suh as the wind veloity w and the veloity of the ground point are doubly primed. merial simulations of measurement sequenes and measurement proessing with relativisti orretions to extrat the horizontal omponent of the wind are disussed in Setion 5. Effets due to spaeraft aeleration are derived in Setion 6 from the point of view of the ECI frame, and these and similar higher-order relativisti orretions are disussed in Setion 7. In Setion 8 an alternative derivation of the frequeny shift is given negleting the relative rotation of the ERF and LRF axes. Relativisti orretions of order v/ and v/ are disussed. Here v7000 m s 1,sov/ Notation. The existene of three or more referene frames in this problem (there is also an instantaneous rest frame for the wind, but suh a frame is not used in Fig. 3. Earth referene frame. This is defined with origin at the ground point below the satterer but fixed on earth s surfae. The wind veloity relative to the surfae is the objet of the lidar measurement and is denoted by w. Quantities suh as the veloity of the spaeraft in this frame are denoted with primes. this paper) results in a notation that is umbersome in some respets. We denote the veloity of the ground point relative to the ECI (see Figs. 1 and ) by, and the veloity of the lidar relative to the ECI by. We assume that in a single measurement the radiation is inident on the aerosol partile for suh a short time that w is onstant. However, during the few milliseonds (ms) of propagation time from the lidar to the satterer and bak to the detetor, the veloity of the detetor relative to the ECI may have hanged. This is fully disussed in Setion 6. In the alulation given in the following setion, we regard the lidar apparatus as being at rest. The aerosol partile is assumed to have veloity w relative to the lidar apparatus.. FREQUENCY SHIFT IN LIDAR REST FRAME In Figure 4, the sattering partile is represented as an irregular objet that moves with veloity w in the x diretion. The wave vetor k of the inident wave has x omponent k os, so that wk os w k. Two wavefronts for the inident wave are shown, the first labeled #1 inident that strikes the partile at some initial time t, and another one labeled # inident, a distane 0 /k away, that will hit the moving satterer a time t later. Figure 5 shows first sattered wavefront #1 refleted as though it were plane, at time t later. During the time t the satterer moves a distane wt in the positive x diretion. At this instant the seond wavefront impinges on the partile; this event would generate a seond baksattered wavefront. Meanwhile the first bak- Table 1. Desription of Frames of Referene Referene Frame Origin Axis Diretions Veloity in ECI Rotation ECI Earth s enter Fixed in spae 0 None LRF Lidar transmitter Fixed in spae None ERF Ground point Fixed in spae None ECEF Earth s enter Fixed in earth 0 Rotating with earth

3 353 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby w w k 1 os 1 f k w. 4 w k 1+ os 1+ k This result applies when the lidar veloity is onstant; a speial ase has been disussed by Gudimetla and Kavaya [4]. Introduing the measured frational frequeny shift / by means of Fig. 4. Two wavefronts are shown traveling toward the sattering partile with speed relative to the LRF. The partile has veloity w. The inident wavefronts are separated in spae by the wavelength 0. The figure is drawn in the LRF at the instant that wavefront #1 strikes the partile; the satterer is suffiiently far from the transmitter that the arriving wavefronts are plane. sattered wave propagates exatly bak toward the transmitter a distane t. The two baksattered wavefronts are separated by the wavelength. The omponent of the distane moved by the sattering partile in the diretion of the inident ray during time t is w os t. The distane between the two inident waves is the wavelength 0 of the inident wave, and from Figure 5 it is seen that 0 t w os t. For the ray sattered bak toward the detetor, the wavelength is the distane between refleted wave #1 and inident wave #. Denoting the wavelength of the sattered photons by, we have t + w os t. Combining the two equations for the inident and sattered wavelength, 0 /f 0 t wt os /f t + wt os. Therefore the ratio of the baksattered frequeny to the inident frequeny is Fig. 5. The first wavefront generates a spherial baksattered wave, shown at the instant the seond wavefront strikes the satterer. The baksattered wave beomes plane as it travels toward the detetor. The satterer moves with veloity w. Suessive wavefronts striking the satterer generate a series of baksattered waves separated in spae by wavelength. 1 3 f 1+f, we may solve for the quantity involving the wind veloity w to obtain 1 5 w k k Thus on the right-hand side of the above equation, only the measured frational frequeny shift is needed in order to determine the wind veloity. If the measurements an be proessed in suh a way that only the frational frequeny shift is involved, then no error in the wind veloity will our resulting from a lak of knowledge of the atual transmitter frequeny. If we denote the unit vetor pointing from transmitter to satterer by n t k/k, this result an be written in the form w n t The term in the denominator is a relativisti orretion sine it is 1 times the numerator. Equation (7) ould be expanded to seond order in the frational frequeny shift, giving w n t f 0 + f + 8 For a spaeraft veloity of 7 km/s, the orretion represented by the seond term in Eq. (8) an ontribute more than 0.15 m/s to the wind veloity. The right-hand side of Eq. (7) is determined by the measurement itself. Thus the omponent of the wind veloity in the diretion of the beam is determined in the LRF. The goal, however, is to obtain the veloity relative to the ground, that is, to find w in the ERF. Simple Galilean transformations lead to appreiable error; instead we need the Lorentz veloity transformations. Equation (7) gives the measured frational frequeny shift in the LRF, and an observable quantity having the frational frequeny shift as a relativisti orretion in the denominator has been derived. Sine the quantity of interest is the wind or aerosol veloity in the earth-fixed

4 Neil Ashby Vol. 4, No. 11/ November 007/J. Opt. So. Am. A 3533 frame, the results must be transformed into the ERF. Additional relativisti effets enter when these transformations are applied. 3. LORENTZ TRANSFORMATIONS IN VECTOR FORM We now prepare to derive a general expression for the wind veloity, not restrited to leading-order relativisti orretions. In this setion we introdue the vetor form of the Lorentz transformations, whih is a ompat way of writing the transformations. This form of the Lorentz transformations an be found in many textbooks [1 3]. A general Lorentz transformation of position an be separated into a part parallel to the relative veloity, whih suffers Lorentz ontration, and a part perpendiular to the relative veloity, whih is unhanged. Sine oordinate differenes are of interest in the derivations that follow, onstant translations of origins are not signifiant. We shall disuss the transformation from ECI to ERF oordinates with parallel axes. Let be the veloity of the ground point in the ERF on earth s surfae, relative to the ECI frame. We assume this will be known aurately in terms of the position of the ground point and the time. Let r be a position vetor in the ECI frame, and denote position vetors and time measurements in the ERF with primes. Then the omponent of r parallel to is r, 9 and this part of the vetor suffers Lorentz ontration, expressed in the transformation equations by multiplying by the fator E 1 1 /. 10 The omponent of r perpendiular to the veloity is r r. 11 Then the Lorentz transformations in vetor form are r r r + E r + t, 1 t Et + r, 13 sine only the omponent of r parallel to suffers Lorentz ontration, and only the omponent of r parallel to the veloity enters the time transformation. Dotting into Eq. (1), r E r + t. The inverses of these transformation equations are r r + E 1 r E t, 14 t Et r, r E r t. 15 We shall also need veloity transformations, obtained by taking differentials of both sides of the position and time transformations. Thus, for example, from Eqs. (15), dr dr + E 1 dr E dt; 16 dt Edt dr, 17 and taking the ratio of orresponding sides of the above two equations, we obtain the vetor form of the veloity transformation equations for v in terms of v and : dr + E 1 dr E dt dr dt v V E Edt dr 1 V E / v V E / v 1 v, 18 where the veloity of an objet measured in the ECI frame is vdr/dt. We also need the inverse of Eq. (18): v 1 V E / v V E / v + 1+ v, 19 giving the veloity v in the ECI frame in terms of the veloity v in the ERF. 4. CONSTANT SPACECRAFT VELOCITY In this setion we give a relativisti treatment of the lidar measurement, assuming that the spaeraft veloity does not hange during the time interval between transmission and detetion of the signal. We assume for simpliity that the sattering event ours at a fixed position r s and time t s in the LRF; that is, that no signifiant time delay ours during the interation between pulse and satterer. Also, we assume that the detetor is effetively at the same point as the lidar transmitter in the LRF, so that the path of the refleted signal lies exatly on top of the transmitted signal path. Then if d t is the displaement

5 3534 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby from transmitter to satterer and d d is the displaement from sattering partile to detetor, we have d t + d d 0. 0 L 1 L, V 1+ L L L, 6 The task is to transform Eq. (7) into the ERF, to obtain a useful expression for the wind veloity w. The proess involves transforming first from LRF to ECI and then from ECI to ERF. When two parallel-axis Lorentz transformations in different diretions are ompounded, the axes of the final frame are no longer neessarily parallel to the axes of the initial frame. This issue is disussed in Setion 9. Suppose that we speify the line of sight (LOS) in the ECI. This is an important ase beause the spaeraft orbit is most likely to be speified in the ECI, and the ground point position and veloity will be known or an be alulated if the ECI time of the lidar shot is known. Our starting point is therefore Eq. (7). The vetor n t from the lidar transmitter to the aerosol partile is presumed known in the ECI system. To obtain an expression for n t in terms of n t, use Eq. (15) with in plae of and r in plae of r. Then, for the vetor displaement between transmitting and sattering events, we have d t r s r t r s r t + L 1 r s r t where L t s t t, L We define the vetor from transmitter to satter by r s r t d t. 3 Then by the onstany of the speed of light, t s t t d t, and we have d t d t + L 1 d t L d t. 4 The magnitude d t of the displaement vetor is obtained by taking the dot produt of d t with itself. Sine there are three terms in Eq. (4), there are basially six terms in the square: d t d t d t d t + L 1 d t d t L d t + L V L d t V L +1 L d t L L V L + L d t. 4 d t 5 There is onsiderable anellation. Then using the identities the expression beomes a perfet square, giving the result d t L d t1 d t d t. 7 Then we may form an expression for the unit vetor n t in the LRF in terms of quantities observed or measured in the ECI: n t + L 1 n t L d t V n t d t L V, 8 L n t L1 where n t d t /d t. Also, it then follows that n t V L / n t 1 V. 9 L n t In the following derivation, the left-hand side of Eq. (8) will be used as an abbreviation for the right-hand side, in whih the quantities are expressed in the ECI frame. We want to express w in terms of w in the ERF. We shall do this in two stages, first transforming with a parallel axis transformation to the ECI frame, and then by a parallel axis transformation to the ERF. The veloity transformation to the ECI is w w + L 1 w L L1 w. 30 Therefore, dotting n t into this expression, we have from Eq. (7) n t w w n t + L 1 L n t L1 w We manipulate this equation so that all terms involving w are isolated as linear terms on the left side. As a first step,

6 Neil Ashby Vol. 4, No. 11/ November 007/J. Opt. So. Am. A 3535 n t t + L 1 w n L n t +1 w 1+ 1, 3 and then moving the term on the right that has w in it to the left, n t t + L 1 w n n t + L V L f To save writing, we introdue the following abbreviation: n t A n t / L + 1 1/ L + V L Then we obtain w A n t On the right side of Eq. (35), the quantities are presumed known from the lidar measurement or from omputations performed in the ECI frame. Also, on the left, the vetor A may be presumed known. A is almost but not quite a unit vetor. Also, although A appears to depend on doubly primed quantities, the appearane of n t is to be regarded as shorthand for the substitution represented in Eq. (8). Inserting Eq. (8) into the definition of A, after some simplifiation we obtain the following expression for A: A n t L 1 n t + V L The next step is to extrat that part of w that depends on w, the desired wind veloity in the ERF. The veloity transformation from Eq. (19) is w w + E 1 w + E E1+ w. 37 Dotting A into this equation, and using Eq. (35), we have w w A + E 1 A + E A n t + E1+ w Again, a term in w ours in the denominator. Multiplying by the fator in the denominator that has w in it, then rearranging so that all terms involving w are on the left, the result is A + E 1 w A E n t + n t A. 39 We introdue an abbreviation for the quantity in square brakets: A + E 1 B A + n. t E Then the result is w B n t A. 41 The main result, Eq. (41), or appropriate approximations to it, should be used in analyzing Doppler lidar data. We negleted spaeraft aeleration in deriving Eq. (41). Obviously numerous relativisti orretions enter the result. As a first hek, let us pass to the nonrelativisti limit, in whih all terms involving 1/ are negleted. Then

7 3536 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby n t n t, A n t, B A n t. We also neglet the term /f in the denominator and use /. We obtain w n t + n t. 4 Thus the theoretial expressions tell us how to remove the veloities and from the measurement so that the omponent of wind veloity w along the LOS is determined. [In Eq. (4) the LOS is speified in the ECI frame.] Sine the spaeraft veloity has by far the largest magnitude of all veloities in the problem, suh subtrations will typially involve the loss of two or three digits of auray. For example, if the wind veloity w were zero, there ould still be a large measured Doppler shift due to the large veloity of the spaeraft relative to the ground. Then Eq. (4) shows that most of the observed Doppler shift will be aneled by the term n t, and the remaining shift will be aneled by the term n t. Equation (4) ould be tested by observing the baksattered ground return signal. Keeping terms of O / only, we may expand Eq. (36): Fig. 6. Possible situation with parallel to the ground, at angle from the uniform wind field. The sanning angle is 30 from the nadir. w n n t t1+ + V L n t n t / + n t + n t / A n n t t1+, 43 n n t t1+ + n t n t V L. 46 B A. 44 The result of substituting these approximations into Eq. (41) is w n t n t n t n t n t n t. 45 The additional terms added to n t inside the parentheses are an effet of relativisti aberration the hange of apparent diretion of a light ray when a transformation between relatively moving referene frames is introdued. Here, the terms n t n t are perpendiular to n t. Thus the first-order relativisti orretions serve to add terms that slightly hange the diretion of the LOS. There are several relativisti orretions in Eq. (45); most of them are of omparable orders of magnitude. On the right side of this equation, there are orretions of order /* to the main term, whih is n t. A typial orretion ontributes about 0. ms 1 to the right side, but the net effet depends on the geometry. Expanding to seond order in powers of / and / gives The largest seond-order terms ontribute orretions to the wind speed of order / * m/s,or rad to the diretion of the line of sight. Suh small orretions are probably negligible for near-earth appliations. In any ase, expansion to seond or higher order gives expressions that are so umbersome that it is probably better to use exat expressions in numerial alulations. Figure 6 shows an example in whih there exists a uniform horizontal wind field and a horizontal relative veloity of the spaeraft relative to the ground. The sanning angle of the lidar is at 30 relative to the nadir Fig. 7. Error in wind veloity resulting from neglet of denominator in last term of Eq. (41). The three urves are labeled by the sanning angle relative to the nadir.

8 Neil Ashby Vol. 4, No. 11/ November 007/J. Opt. So. Am. A 3537 in the diagram. Omission of the relativisti orretions in the denominator in Eq. (45) results in systemati errors in determining the wind speed that are plotted in Fig. 7 for various sanning angles. In this example 300 ms 1, and the altitude of the lidar was 400 km. To obtain a better idea of the importane of relativisti orretions, we have developed a numerial simulation of the measurement proess involving an earth-orbiting lidar apparatus. This will be explained in the next setion and some examples will be disussed. 5. SIMULATIONS: EXAMPLES In this setion we desribe numerial simulations using Eq. (41). These examples illustrate how the theory is to be applied and also indiate how large the relativisti orretions an be. The sattering partile is assumed to be on earth s surfae, at the hosen observation point of latitude/longitude N, 105 W. At this point the wind is assumed to be onstant, w m/s E, m/s N. We assume the spaeraft is in a Keplerian orbit haraterized by the following orbital parameters, from whih the position and veloity in the ECI frame an be derived when the time is speified: R e m, earth radius, h m, nominal altitude, a R e + h m, orbit semimajor axis, 49 E e sin E GM a 3 t t p, 56 where GM m 3 /s is the produt of the Newtonian gravitational onstant and the earth s mass. Then the true anomaly f is found from os f os E e 1 e os E. 57 The radius of the spaeraft is obtained from r a1 e os E. 58 The ECI Cartesian oordinates of the spaeraft are then found from elementary trigonometry: x ros osf + os I sin sinf +, y rsin osf + + os I os sinf +, z r sin I sinf Assume that a pulse from the lidar apparatus starts to the ground point at tt t s exatly. This time is speified in the ECI frame. The position and veloity of the lidar at this moment is obtained by solving the Keplerian equations of motion, and is found to be r L , , m, , , m/s. 61 e eentriity, 50 To find the time t s when the pulse arrives at the ground point, one uses onstany of the speed of light: rad, altitude of perigee, rad, angle of asending line of nodes, i 60.00, orbit inlination, t p s, time of perigee passage A. First Lidar Measurement Given the time of perigee passage above, and assuming the Greenwih meridian of earth rotates in the ECI referene frame with angular veloity e rad/s, the azimuthal angle of the ground point as a funtion of time t (in the ECI referene frame) is t 105 /180 + e t. 55 The fators /180 in the first term of the above equation just onvert the angle to radians. In a similar manner, any point with given latitude and longitude on earth s surfae an be found in the ECI referene frame. Also, at any given time, with the Keplerian orbital parameters listed above, the ECI position of the lidar transmitter an be found by solving the Keplerian equations of motion [5]. First one solves for the eentri anomaly E in the equation t s t t + 1 r st s r t t t, 6 where r s t s and r t t t are the positions of satterer and transmitter at the sattering and transmission events, respetively. An estimate of the time of arrival t s is then used to reompute the ground point position in the ECI frame; then the distane between the ground point and the transmission event is realulated, giving a better estimate of the arrival time. This proess onverges after three or four steps and gives for the time of arrival t s s. 63 The ground point position and veloity at this instant are r E , , m, , , m/s. 65 (The z axis is hosen parallel to earth s rotation axis.) These data allow us to ompute the vetor d t and to transform it into the LRF: d t , , m, 66

9 3538 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby d t m. 67 Using the relativisti omposition law for veloities, we first transform the given ERF wind veloities to the ECI, and then to the ERF. This is so we an alulate the expeted frational frequeny shift. The veloity of the wind relative to the lidar apparatus will be w , , m/s. 68 With these simulated values we an ompute the frational frequeny shift that should be observed from Eq. (4): and the quantity 1+ 1 f , The vetor A defined above is A , , Then the vetor B may be omputed: B , , f + n t m/s. 74 Thus the lidar Doppler shift measurement is large beause the spaeraft veloity is large. Most of the shift must be subtrated out to aount for veloities of the spaeraft and ground point in order to measure the wind. B. Seond Lidar Measurement Let a seond measurement be made exatly 150 s later. For this measurement the lidar is from the ground point zenith, whereas for the first measurement the lidar was 59.8 from the zenith. (These onditions were arbitrarily hosen and do not orrespond to a onstant sanning angle.) The physial onfiguration of the two lidar measurements is drawn to sale in Fig. 8; there it is seen that the first laser shot ours as the lidar is approahing the satterer, and the seond shot ours as the lidar is reeding. During the time interval between the shots, the satterer moves due to the ombination of wind and earth rotation. Relativisti effets are too small to show up in Fig. 8. For the seond laser shot, the measured frational frequeny shift is f , 75 and the result of evaluating the left side of Eq. (41) is m/ s, while evaluating the right-hand side gives m/ s. Again, the agreement is extremely good. These two simulated measurements give the wind veloity relative to the ground point. For the first measure- These two vetors turn out to be essentially equal; this is beause the speed of the ground point is very small ompared with the speed of light. In a first approximation one an in fat neglet the differene between B and A. Resolving the vetor B into eastward, northward, and vertial omponents, we quote only the horizontal omponents sine the wind is assumed to be horizontal: B E, N. 73 These results an now be used to test Eq. (41). Using the input values of the wind at the ground point gives for the left-hand side of Eq. (41) m/ s, while evaluating the right-hand side gives m/ s. The agreement is extremely good, whih gives one onfidene that the simulation properly represents the theory. It is of interest to examine the onsequenes of making approximations leading to Eq. (4). The LOS vetor is defined in the ECI referene frame. The left-hand side of Eq. (4) has the nonrelativisti value m/ s, while the right-hand side evaluates to m/ s. The differene is.4 parts in 10 4 ; there has been a loss of auray from the subtration of two large terms on the right: Fig. 8. Vetors for the simulation in Setion 5, viewed from the ECI referene frame. Latitude longitude grid lines are drawn at the time of the initial measurement 0.5 apart. Displaement of the ground point between the measurements is due to earth rotation and is along a parallel of latitude.

10 Neil Ashby Vol. 4, No. 11/ November 007/J. Opt. So. Am. A 3539 ment, the vetor B has been given above. Assuming the wind has eastward and northward omponents w e,w n gives us one equation: w e w n m/s. 76 t d t t + gt d t t, and the detetor position will hange to r L t d r L t t + t t t d t t + 1 gt d t t, 81 8 For the seond measurement, evaluation of the vetor B gives B , , , 77 and resolving this into eastward, northward, and vertial omponents aounting for the additional rotation of earth, B E, N. 78 Assuming that the wind remains onstant in between the measurements gives w e w n where g is the aeleration due to earth s gravity at the spaeraft. We let n t and n d be unit vetors pointing along the path of the transmitted and sattered rays, respetively. The interation time of the radiation with the satterer is assumed to be so short that the positions of the transmission event, sattering event, and detetion event an be desribed as happening at r t t t, r s t s, and r d t d, respetively. We neglet atmospheri refration or dispersion effets and onfine this disussion to relativisti effets. For the transmitted radiation, we onsider two suessive wavefronts that start at t t and t t +dt t and arrive at the satterer at times t s and t s +dt s, respetively. Then from the onstany of the speed of light we have m/s. 79 t s t t 1 r st s r t t t, 83 Solving the two simultaneous equations (76) and (79) for the wind veloity gives w e,w n , m/s, 80 whih agrees very well with the assumed wind veloity omponents w e,w n 50,47 m/s. If the wind has a vertial omponent, a third measurement is needed to determine all three wind omponents. 6. ACCELERATED SPACECRAFT We reanalyze the measurement from the point of view of the ECI frame, but we inlude aeleration of the spaeraft and the effet of earth s gravitational potential on the transmitted and deteted signal frequenies. The spaeraft s aeleration g in earth s gravity field hanges the spaeraft position and veloity during the light travel time, so the deteted frequeny will be affeted. For a light travel time of 8 ms, the spaeraft veloity hange will be about 0.07 ms 1. If not aounted for the measured wind veloity ould be in error by about half this, or ms 1. To treat this effet it is most onvenient to analyze the measurement in the ECI frame, allowing for different spaeraft veloities at the transmission and detetion events. The alulation will be done without making any approximations in the relativisti transformations; then ontributions of various orders will be disussed. Let t t be the time of transmission of a wavefront from the lidar apparatus, and suppose it arrives at the satterer at time t s. At the instant of transmission the lidar has veloity t t and position r L t t. The pulse is immediately sattered bak into a detetor, whih is assumed to be oinident with the transmitter (monostati lidar). The arrival time of the sattered wavefront at the detetor is denoted by t d. During the propagation delay time t d t t the detetor veloity will hange to and differentiating to obtain expressions for the differentials dt s,dt t, we have dt s dt t 1 r s t s r t t t dr s t s dr t t t. 84 r s t s r t t t The unit vetor along the LOS of the transmitted ray is n t r st s r t t t r s t s r t t t. 85 Also, the position inrements are related to the veloities Thus dr s dr s dt s dt s wdt s, dr t t t dt t. 86 dt s dt t n t wdt s t t dt t /, 87 and this may be rearranged to give dt n t w s1 dt n t t t t1. 88 The oordinate time interval dt t an be related to the orresponding proper time d t on a standard lok at the transmitter, and hene to the proper frequeny of the transmitted signal, by using the ordinary salar invariant of the gravitational field. We shall suppose that the proper time interval d t orresponds to the emission of two suessive wavefronts by the transmitter so that d t 1/. The proper time is given to suffiient auray by d t 1+ r t dt t dx +dy +dz, 89

11 3540 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby 1+ r t t t dt t, 90 where r t is the Newtonian gravitational potential of earth at the transmission event. Thus d t 1+ r t t t dt t. 91 For the emission of two suessive wavefronts, this time is the reiproal of the frequeny, and so dt t 1 1+ r t and therefore using Eq. (88) dt s1 n t w V, 9 Lt t 1 n t t t 1+ r t t t. 93 We perform a similar analysis for the suessively sattered wavefronts oming bak to the detetor and use the result to eliminate dt s. From onstany of the speed of light, t d t s 1 r dt d r s t s, 94 and differentiating to obtain expressions for the differentials dt d,dt s, we have dt d dt s 1 r d t d r s t s dr d t d dr s t s. 95 r d t d r s t s The unit vetor along the LOS of the sattered ray is n d r dt d r s t s r d t d r s t s, 96 so inserting veloities of wind and lidar detetor, substituting in the definition of n d and rearranging, we obtain dt n d w s1 dt n d t d d1. 97 The oordinate time interval dt d at the detetor is related to the proper time inrement at the detetor, and hene to the deteted frequeny f, by and therefore d d 1 f 1+r d dt d 1 f 1+ r d Substituting this into Eq. (97), t d dt d, 98 V. 99 Lt d dt s1 n d w 1 n d t d f 1+ r d t d. 100 Dividing Eq. (100) into Eq. (93), aneling dt s, and rearranging, we obtain the following expression for the ratio of deteted to transmitted frequeny in the LRF: r t 1+ f t t 1+ r d t d 1 n t w/ 1 n d w/ 1 n d t d / 1 n t t t /. 101 This relationship ontains all the information available to determine the wind veloity. It allows for the possibility that t d and r L t d differ from t t and r L t t, respetively. The desired wind veloity w will be introdued by making a Lorentz transformation. Let us first onsider the ratio of square roots that ours on the right side of Eq. (101). If the spaeraft orbit were perfetly irular, then to a high degree of approximation the spaeraft speed and altitude would not hange and the square root fators would anel. Even if the orbit is slightly eentri, the eentriity e is expeted to be rather small, say e0.01. Aounting for the hanges due to earth s gravity as in Eqs. (81) and (8), the ratio is approximately 1+ r t 1+ r d t t t d 1+ t d t t g, 10 and the last dot produt will bring in a fator e. Thus this term is small not only beause the propagation delay interval is small but also beause there is another fator of e and there are already fators in the denominator. The net effet is that under almost all irumstanes the square root ratio differs from unity by less than and an be negleted. We shall therefore not onsider it further in this paper. The expression for the measured frequeny redues to f 1 n t w/ 1 n d t d / 1 n d w/ 1 n t t t /. 103 If Eq. (103) is transformed to the LRF, assuming the veloity is onstant, the result an be shown to agree with Eq. (4). Thus Eq. (103) provides an alternative formulation, in terms of quantities speified in the ECI frame, from whih the wind veloity may be derived. With Eq. (103) we an estimate the prinipal ontribution of spaeraft aeleration to the measurement. Introdue the frational frequeny shift by means of Eq. (5) and expand the right-hand side of Eq. (103), keeping only terms of first order in w/ and /. Then

12 Neil Ashby Vol. 4, No. 11/ November 007/J. Opt. So. Am. A n t w/ + n d w/ n d t d / + n t t t /. 104 To a very good approximation, n d n t, and w +w. Also, the aeleration auses a hange in veloity given by Eq. (81). With a little rearrangement we obtain w n t + n t + 1 n t gt d t t. 105 For reasonable distanes of 160 km between transmitter and target, the propagation delay is about t d t t s, and g9 ms. So the aeleration term in Eq. (105) ould ontribute to the wind speed as muh as ms m s 1, 106 but this ould be further redued if the angle between n t and g is large. We have already disussed the approximate anellations of the two main terms on the right side of Eq. (105). The next step in the alulation is to transform Eq. (103) so that the wind veloity w in the ERF appears expliitly, without making further approximations. The alulation is tedious, and to omplete it with a minimum number of steps we shall use the Lorentz transformations in vetor form. At any stage in the alulation one may hoose whether to transform the LOS unit vetors n t and n d into some other referene frame. We shall use only parallel axis transformations from the ECI frame to one or the other of the LRF and ERF. With this in mind, the starting point is the expression, Eq. (103). To save writing, we introdue the abbreviation F f 1 n t t t / 1 n d t d /. 107 This quantity inludes the measured frequeny of the baksattered radiation and Doppler shift fators that an be alulated when the veloities and the LOSs are known in the ECI. Then Eq. (103) beomes F 1 n t w/ 1 n d w/, 108 and solving for the terms involving w, we have w n t Fn d 1 F. 109 We next introdue the desired wind veloity w by substituting the veloity transformation, Eq. (19), into Eq. (109). Then several dot produts involving w appear. The result an be written 1 F w E1+ w n t Fn d + E 1 w n t Fn d / + E n t Fn d. 110 We ollet all terms involving w on the left: w n t Fn d E 1 F + E 1 Fn d E1 F E n t Fn d. n t 111 Spaeraft Aeleration. Let us again examine the effet of spaeraft aeleration by assuming the wind veloity is very small. The right side of Eq. (111) redues to 1 f 1 n t t t / + n d t d / n t n d + n t t t n d V d t d n t + n d Inserting the spaeraft aeleration, from Eqs. (81) and (8), there will be a ontribution to the wind speed of about n d gt d t t /.08 m s 1, 113 as disussed in onnetion with Eq. (105). A fator of 1/ arises from the left side of the equation, so the result agrees with Eq. (105). In the rest of this paper we shall ignore spaeraft aeleration and onentrate on other relativisti orretions. 7. ALTERNATIVE FORMS FOR RELATIVISTIC CORRECTIONS Equation (111) is an alternative form of the result inluding relativisti orretions, with onstant spaeraft veloity. This equation an be redued to Eq. (41) by dividing eah term in Eq. (111) by E L 1 n t /1 +/. For example, the following identities an be proved: L 1 n t 1+1 f 0 A n t Fn d, L 1 n t n t F. 115 In Eq. (111), the quantity into whih w is dotted, as well as the quantity on the right side of Eq. (111), are expressed in terms of the measured frequeny ratio f / in the LRF and veloities speified in the ECI frame. Various ases of interest may be disussed; for example, relativisti orretions may be obtained by expanding Eq. (111) to the desired order in w/, /, or /. Heneforth we shall neglet spaeraft aeleration and assume that In this ase t d t t. 116

13 354 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby d d + d t t d t t, 117 and using t d t t d d +d t, one an exatly solve for d d and d d in terms of d t. The solutions are useful, so we quote them here. First, alulating the square of d d gives rise to a quadrati equation for d d. By an argument similar to that leading to Eq. (7), the physial solution is 1 n t / + V L / d d d t 1 V. 118 L / Then putting this bak in to Eq. (117) gives d d d t + d t 1 n t / 1 V. 119 L / If all first-order orretions are negleted in Eq. (111), F 1+/, and the equation an be redued to w n t + n t n t, 10 whih is equivalent to Eq. (4). To obtain first-order orretions, we first need to expand n d in terms of other quantities. This omes from Eqs. (118) and (119) and is n d d d d d n t 1+ n t / + /. 11 Expanding Eq. (111) to leading order in w/, /, and /, the following result is obtained: w + n t 1+ n t / / Equation (1) agrees with Eq. (45) to this order. Sine Eq. (111) has the same physial ontent as Eq. (41), expansions to higher order will also agree. 8. CORRECTIONS NEGLECTING RELATIVISTIC ROTATION OF AXES In this setion we investigate relativisti orretions under the assumption that a diret transformation between LRF and ERF an be made, whih means negleting the differene in orientation between the two sets of axes. Equation (18) allows us to introdue the veloity of the LRF relative to the point on the ground. Let the veloity of the lidar apparatus as measured in the ECI frame be. Substituting this for v in Eq. (18) gives 1 VE / / V E Sine the positions of spaeraft and earth point are presumed known, all quantities appearing on the right side of this equation are known. We may thus proeed to transform the observable quantity, Eq. (7), totheerf. In the following equations, we make diret Lorentz transformations (reall quantities with primes are measured in the ERF); r r r + L 1 + L t 14 r t L t +, 15 where L is a funtion of, 1 L 1 /. 16 Transformation to the ERF. We derive an equation satisfied by the wind veloity in the ERF. To do this we transform the observable diretly from the LRF into the ERF, using the ECI frame only as an intermediary for alulation of the relative veloity of the two frames. Using the Lorentz transformations, for the transmission event we have r t r t r t + L 1 + L t t, 17 and for the arrival event of the pulse at the ground point, r s r s r s + L 1 + L t s. 18 Subtrating the first of these equations from the seond, we have an expression for the vetor d t from the transmission event to the arrival event in the ERF: d t d t r s r t d t + L 1 + L t s t t. 19 But t t s t d t in the LRF, so we find for the path vetors d t d t d t + L 1 + L d t. 130 We an find the magnitude of d t by dotting it into itself. The result is d t d t L d t 1+, 131 d t and the inverses are

14 Neil Ashby Vol. 4, No. 11/ November 007/J. Opt. So. Am. A 3543 d t d t d t + L 1 L d t, 13 d t d t L d t d t We leave the frational frequeny differene as measured in the LRF unhanged but transform the expression on the left side of Eq. (7). Then we need the quantity where we have double primes on d t and w as in Eq. (7) to remind us that these quantities refer to measurements made in the LRF. The veloity w of the wind in the ERF is obtained by diret Lorentz transformation: w + L 1 w/ L w. 135 w L 1 w d t, 134 d t w d t d t 1+ 1 We may now form the observable, Eq. (134): w V w + L 1 L L d t d t + L 1 L V L L w d t d t 1 1 d t d t. 136 The objet of the following derivation is to simplify the above expression as muh as possible, isolating all quantities involving w in the numerator of one side of the equation. Note that everything is presumed known in this equation exept w. Putting some of the fators that are in the denominator of Eq. (136) on the left, working out all the terms in the dot produt and simplifying, we obtain f L 0 w f d t + w d t w V d t d t 1 d t L d L + L t d t 1 d t. 137 The terms involving w an be put on one side of the equation: w n t / L w V n t 1 L f V L n t n t 1 f 0, 138 where n t d t /d t. The quantities appearing in this expression should be known, either from measurement or from knowledge of the spaeraft orbit and ground position given the time of the measurement, exept the wind veloity w in the ERF. Let us introdue a vetor D t by means of n t V D t L 1 n t L f Then the result for the wind veloity in the ERF involves a dot produt: V L n t w D t n t 1 f The result gives the omponent of w, the wind veloity, in the ERF, along D t. D t is not quite a unit vetor. It also depends on the measured frequeny shift, so it is a ompliated but presumably known objet. Relativisti orretions of all orders are ontained in this result. For example, there are several relativisti orretions ontained in the expression for the relative veloity, given in Eq. (13). However, beause the diret transformation has ignored the nonparallelism of the ERF and LRF axes, the results annot be expeted to be aurate to O. Expanding Eq. (140) to first order in /, the result an be shown to agree with Eq. (45).

15 3544 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby 9. NONPARALLELISM OF AXES IN ERF AND LRF The theory presented in the previous setions treated the LRF, the ECI frame, and the ERF as though they were parallel. This is not stritly true, sine it is well known that when two nonollinear boosts are involved the net transformation involves a boost and a rotation. In the present ase, the veloities and are generally not in the same diretion. Therefore the axes of the LRF and the ERF are generally not parallel. Equation (13) gives, the veloity of the LRF relative to the ERF. The veloity of the ERF as measured in the LRF would be of the same form as Eq. (13) with and interhanged. We all this veloity, and find 1 VL / / V L It is easy to see that and have the same magnitude, but they do not point in opposite diretions in the vetor sense. If we let be the angle required to rotate into, and let Nˆ be a unit vetor along the rotation axis, then Nˆ sin. 14 After some algebra, and reduing the expression to its lowest-order ontribution by expanding in powers of / and /, wefind Nˆ sin. 143 This rotation angle is typially only a few parts in This orretion is of order / or / smaller than the first-order relativisti orretions disussed earlier. It may be negligible in the urrent appliation. Using the transformations developed in this paper, and negleting the differene in diretion of and, from Eq. (140) one may transform diretly from the ERF to the LRF and expand to seond order and obtain n t w n t + 1 n t V 1+ L n t V +1+ L n t. 144 Fig. 9. The relative veloities and are not antiparallel. The diagram is drawn in the plane formed by and. On the other hand, one an transform diretly from the LRF to the ERF using Eq. (7), assuming the veloity of the LRF relative to the ERF is, expand to seond order and obtain n t w n t 1+ n t V 1+ + E n t V 1+ E n t. 145 Equations (144) and (145) appear at first to be idential, assuming ; however, suh an identity ignores nonparallelism of axes of the ERF and LRF. Neither Eq. (144) nor Eq. (145) agrees with results derived earlier [Eqs. (41), (111), and (140)] that were derived without assuming the axes of the ERF and LRF were parallel; thus they are inorret and should not be used beyond first order in v/, where v is any veloity in the problem. Eqs. (144) and (145) provide alternative formulations, depending on different veloity variables, that are valid only to first order in v/. The differene of orientation of the axes in the LRF and in the ERF is illustrated in Fig. 9, in whih the vetors and are drawn in the plane of the paper. These two vetors would be antiparallel if the referene frame axes were parallel. The angle between and is expressed approximately in Eq. (143) in terms of a vetor rotation angle /, whih would be normal to the plane of the paper. 10. ON-BOARD GPS RECEIVER If a GPS reeiver is plaed on the spaeraft, then the lidar position, veloity, and time may be available diretly from the reeiver. Ordinarily the GPS reeiver will provide position and time in earth-entered earth-fixed (ECEF) oordinates. (The origins of the ECEF frame and ERF are at earth s enter and at the ground point, respetively.) The ECEF frame is rotating with respet to the ECI frame; the rotation introdues some new relativisti

16 Neil Ashby Vol. 4, No. 11/ November 007/J. Opt. So. Am. A 3545 effets. We show here how to transform measurements made by the GPS reeiver to the ECI frame so that the previous analysis then applies. Fortunately the rotational speed is so slow that only effets of first order in v/ need to be onsidered. The ECEF frame must be distinguished from the ERF that we have introdued beause of the time delay between transmission and detetion. In the noninertial ECEF frame light does not travel in all diretions with a unique speed. In the present setion we shall plae primes on quantities measured in the rotating frame, keeping in mind that it is not an inertial frame. Let us imagine freezing the axes of the ECEF frame at the instant a signal pulse is transmitted, and identifying the diretions of these axes with those of an ECI frame. GPS time is effetively the same as the oordinate time t in the ECI frame, so we do not have to introdue any transformations for the time. For the transmission event, then r t r t. 146 The sattering event will our at time t s and at position r s. The axes of the ECEF will have rotated through an angle E t s t t, where E is the WGS-84 value of the angular rotation rate of earth. Thus the positions of the sattering event in the frozen ECI frame, and in the ECEF, are related by a rotation: xs y s s z 1 E t s t t 0 E t s t t 1 0 y s 0 0 1xs z. s 147 Here we shall work out the impliations of this rotation only to first order in the small parameter E d/ 10 7, where L7000 km is the largest reasonable distane in the problem. It is then straightforward to ompute the ECI distane in terms of the ECEF distane: r t r s r t r s +t s t t r t r s. 148 The ross produt in the orretion term in Eq. (148) an be interpreted as an area: A 1 r t r s, 149 where A is the vetor area swept out by a vetor from the earth s rotation axis to the signal pulse as the pulse travels along the propagation path. Thus r t r s r t r s +4t s t t A, and taking a square root, 150 r t r s r t r s + t s t t A. 151 r t r s The time interval between transmission and sattering is t s t t 1 r t r s 1 r t r s + E A. 15 Then the expression for the distane an be simplified: r t r s r t r s + E A. 153 The orretion term in Eq. (15) is alled the Sagna orretion. In the present appliation it is only a few nanoseonds. The orretion term in Eq. (153) is only a few meters. The veloity of the lidar in the ECI and ECEF frames are related approximately by + r t. 154 The relativisti omposition of veloities is fortunately not needed here beause the speed of a fixed point in the ECEF system is so small ompared with, provided the point is not too far from the earth s rotation axis. Equations (15) (154) an be used to transform information obtained from an onboard GPS reeiver to the ECI frame, in whih the previous analysis is valid. Attempting to express the equations entirely in terms of quantities measured by the GPS reeiver leads unprodutively to a large amount of algebra. 11. SUMMARY In this paper derivations of the Doppler shift measured by a lidar apparatus, when the inident beam is baksattered from an aerosol partile arried along in the wind, have been presented. One of these derivations was from the point of view of the rest frame of the lidar [see Setion 4, Eq. (41)], and the other was from the point of view of the ECI frame and permitted the alulation of gravitational effets [see Setion 6, Eq. (101)]. The results agree when aeleration and gravitational frequeny shifts are negleted, and show how relativisti orretions enter into the fundamental equation for the wind veloity in the lidar rest frame. Expressions for the wind veloity in the rest frame of the ground point, the earth-fixed point underneath the wind, were derived. Several ways of defining the LOS vetor were disussed, and relationships between them were derived. Simulations of the lidar measurement were developed and examples were given, demonstrating the onsisteny of the approah. Relativisti orretions were derived; for most purposes only the leading relativisti orretions need be onsidered in onverting lidar Doppler measurements to wind veloities, but the theory given here [as in Eq. (101)] should be good to all orders in the small parameters v/, where v is any veloity in the problem. ACKNOWLEDGMENT I am grateful to Mihael Kavaya for suggesting this investigation and for many helpful disussions.