1. Introduction

The National Oceanic and Atmospheric Administration (NOAA)/Earth System Research Laboratory (ESRL) has developed two coherent Doppler lidars to study atmospheric boundary layer dynamics (see Grund et al. 2001; Brewer et al. 1998). Typically these systems perform low–elevation angle scans to determine wind speed and direction and to quantify turbulence with high temporal and vertical resolution. This is accomplished by operating with a high pulse-repetition frequency and controlling the direction of the lidar beam with a hemispheric scanner (see Banta et al. 2002; Banta et al. 2006). The NOAA/ESRL Doppler lidars use coherent detection to measure the radial velocity component of atmospheric scatterers (typically aerosol) relative to the velocity of the lidar; “radial” refers to the direction along the instantaneous lidar beam. Since 1998, the systems have performed wind measurements from seagoing research vessels during eight field studies described in Bretherton et al. (2004), Wulfmeyer and Janjic (2005), Rauber et al. (2007), and Angevine et al. (2006). Twenty-foot seagoing cargo containers (seatainers) house the systems and serve as mobile laboratories. The seatainers were rigidly mounted to the ships. To compensate for ship motion, a motion compensation system performs the following two functions: it actively stabilizes the pointing of the lidar beam and removes the effect of the ship motion from the Doppler wind measurement. The motion compensation system is applicable to the lidar mounted on any moving platform; we will refer to the platform as the “ship” in this work. To study boundary layer dynamics, the atmospheric velocity relative to an earth-fixed coordinate system is desired. The motion compensation system uses the global positioning system (GPS) to define the earth-fixed GPS coordinate system. To obtain radial atmospheric velocity components in GPS coordinates, correction of the Doppler frequency shift for the ship’s motion relative to the GPS coordinate system is required.

To illustrate the effect of ship motion on the Doppler measurement, we will focus on the NOAA/ESRL miniature master-oscillator power-amplifier (mini-MOPA) Doppler lidar for this paper. A simplified block diagram of the mini-MOPA Doppler lidar is shown in Fig. 1. The lidar consists of a continuous-wave (CW) carbon dioxide (CO₂) laser that provides both the local oscillator (LO) reference beam for heterodyne detection as well as the transmitted beam; that is, it is a “monostatic” lidar. The laser operates at a wavelength of 9.4 μm. Using the MOPA configuration, amplitude modulation is applied to the CW beam to chop out pulses. The pulse width produced by the modulation is adjustable. We typically operate with a pulse width of 400 ns (60 m) when staring at zenith and 1 μs (150 m) when scanning. Each pulsed beam passes through a CO₂ optical amplifier to increase the pulse energy to 1 mJ. In Fig. 1, the black and green beam paths are CW, the green path is LO, and the red and blue beam paths are pulsed beams. A polarizing beam splitter and quarter-wave plate are used as a transmit–receive (TR) switch. This allows common beam-steering optics to be used for both the transmitted and received beams. The transmitted beam is circularly polarized; its diameter is 20 cm, and it has a diffraction-limited divergence of approximately 50 μrad. The lidar operates with a pulse-repetition frequency of 200–300 Hz. The received beam (shown in blue in Fig. 1) is caused by Mie scattering of the transmitted beam from aerosol and water droplets. The received beam passes through the hemispheric scanner and TR switch and is combined with the LO reference beam and is focused on a photo detector cooled by liquid nitrogen. Using heterodyne detection, the photo detector responds to the frequency difference between the reference beam (LO) and the atmospheric return. That frequency difference is the measured Doppler shift caused by the relative motion of atmospheric scatterers and the lidar system.

The transmitted beam is directed into the atmosphere using the hemispheric scanner that contains two mirrors (see Fig. 1). Hereafter, the hemispheric scanner is referred to as the “scanner.” The scanner allows the lidar beam to probe the hemisphere above the seatainer by means of the “azimuth rotation” and “elevation rotation” shown in Fig. 1. The elevation mirror is the final mirror before output of the beam to the atmosphere. To compensate for ship motion, such that the transmitted beam is pointed at zenith (called “zenith stare mode”), the azimuth bearing is fixed while the elevation mirror is rotated around both its elevation and zenith tilt axes (see Fig. 1). To scan off zenith while compensating the beam pointing for ship motion, the zenith tilt is fixed and the scanner moves with azimuth and elevation rotation.

Figure 2 shows the 20-ft seatainer containing the lidar. The seatainer is shown mounted on the bow of the Research Vessel (R/V) Seward Johnson during the Rain in Cumulus over the Ocean (RICO) experiment during January 2005. The hemispheric scanner is shown mounted on the roof of the seatainer, as are four GPS antennas that are part of a JAVAD JNSGyro-4T attitude sensor used by the lidar’s motion compensation system. The lidar’s motion compensation system Cartesian coordinate system is shown in black in Fig. 2; the x and y axes are fixed in the plane of the antennas, and the z axis is perpendicular to that plane. Another motion detection system aboard the R/V Seward Johnson is the Position and Orientation System for Marine Vessels (POS MV). Cartesian axes of the POS MV’s coordinate system are fixed relative to the ship and are oriented forward, starboard, and keelward, in that order. From Fig. 2, the lidar’s x and y axes are not aligned with the POS MV’s forward and starboard axes, although the lidar’s z axis is keelward to excellent approximation.

The lidar has many more optical components than are shown in the block diagram in Fig. 1. The accelerations of the ship cause the lidar to be a noninertial reference frame such that the various optical components are in motion relative to one another as observed from either an inertial or a GPS coordinate system. An inertial reference frame is, by definition, not accelerating (or rotating). Inertial reference frames are used in the derivations below because electromagnetic waves move in straight lines in inertial reference frames; relativistic effects need not be and are not considered below. GPS coordinates are not inertial because they rotate with the earth. The optical components are mounted on a frame that can be treated as a rigid body such that the relative velocities of optical elements are constrained by rigid-body equations. Likewise, the hemispheric scanner is a rigid body constrained relative to the seatainer.

Ship motion necessitates correction of the measured Doppler frequency shift in order to determine the atmosphere’s velocity in the earth-fixed GPS coordinate system. What velocities of which optical elements must be used to determine the correction? The derivation below uses the rigid-body constraints to demonstrate that it is only the velocity of the scanner’s final mirror relative to GPS coordinates that determines the correction. The final scanner mirror is the last mirror before the beam exits to the atmosphere, that is, the “elevation” mirror in Fig. 1. The motion correction is illustrated using data from the RICO experiment. In the derivation below, it suffices to use several mirrors at arbitrary positions rather than to analyze the specific positions of the multitude of mirrors and beam splitters within the lidar. That level of abstraction allows our results to be easily applied to other remote sensing systems on any moving platform. The derivation is the main result of this paper, but a reader may desire to read the summary section first.

Comparison with previous publications, such as Edson et al. (1998) and Schulz et al. (2005), shows variations of the formulation that arise because of different definitions of coordinate systems, angular rates, and Euler angles. Much confusion could result unless such definitions are clearly stated. A particular distinction is made by Edson et al. (1998) between gyro-stabilized systems and strapped-down systems. The lidar’s JAVAD and the ship’s POS MV are the strapped-down type. The three angular rates of body motion relative to GPS coordinates form a vector. The components of the angular-rate vector are given by the lidar’s JAVAD system in the lidar’s coordinate system (black in Fig. 2) and by the ship’s POS MV system in the POS MV coordinate system (forward, starboard, keelward). That differs from formulations stated by Edson et al. (1998) and Schulz et al. (2005), which use components of the angular-rate vector in earth-fixed GPS coordinates. Shipborne phased-array Doppler radars have used electronic phasing of the antenna and ship motion measurements to retrieve wind profiles (see Law et al. 2002).

Airborne Doppler radars require correction for aircraft motion. There is substantial literature on that topic (Lee et al. 2003; Bosart et al. 2002; Heymsfield et al. 1996; Lee et al. 1994; Testud et al. 1995). Those studies do not consider the difference between the velocity of the phase center of the radar antenna and the velocity at the location where motion is measured by an inertial navigation unit. That type of velocity difference, considered by Edson et al. (1998) and Schulz et al. (2005), is one essential aspect of the present work, and it is important here because of the distance between where the motion compensation system reports velocity and the positions of optical elements. Another essential aspect of the present report is the fact that the NOAA/ESRL lidars are spatially extended systems such that during shipborne application the optical elements have different velocities. In the last section, we return to a comparison of our derivation with that of Edson et al. (1998), and the impact of our work and how it enhances the capability to address science problems relative to previous efforts.

2. Doppler shift resulting from reflection from a moving mirror

Consider the Doppler shift resulting from reflection from a flat moving mirror. The following applies to observations from an inertial coordinate system, and nonrelativistic relationships apply. Let k_i be the wave vector of light incident on the mirror with ω_i its frequency, and let k_r be the wave vector of light reflected from the mirror with ω_r its frequency. Choose a point on the mirror; at time t, its position is x(t). The incident and reflected waves have the same phase at x(t). For example, a crest of the incident wave causes a crest of the reflected wave, etc. Therefore,

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Differentiating with respect to time, we have

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Observed from an inertial reference frame, k_i and ω_i are constant. Approximating the mirror velocity as constant, that is, u (t) ≡ dx/dt, both k_r and ω_r are also approximately constant such that

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3. Doppler shift resulting from a sequence of mirrors constrained to a rigid body

Consider the light propagating from a source to mirror 1, to mirror 2, and then to mirror 3, etc. For simplicity, we use the inertial reference frame that is moving at the same velocity as the light source at an arbitrary time t; for brevity, we call it the source’s reference frame. Of course, at times previous and subsequent to time t, this source’s reference frame is not moving with the velocity of the source because the lidar’s source is accelerated by ship motion and the earth’s rotation. Throughout, mention of an inertial reference frame at time t is based on the approximation that the instrument’s translatory motion and change of orientation are negligible during the time required for light to travel from the source to the scatterers and back to the detector. The wave vector of the light propagating from the source to mirror 1 is k_sour, and that light has frequency ω_sour; the wave vector of the light propagating from mirror 1 to mirror 2 is k₁, and that light has frequency ω₁, and so on. Relative to the source at time t, mirrors 1, 2, and 3 have velocities u₁, u₂, u₃, respectively. Repeated application of (1) gives

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Adding these equations gives the frequency difference between the light at the source and the output light after reflection from mirror 3 as follows:

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The rigid-body constraint is that the velocity of any given point r on the rigid body relative to the velocity of the source is

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where Ω is the angular-rotation-rate vector of the rigid body at time t, and r is the position vector from the source to the given point. Let c denote the speed of light. To excellent approximation in the limit |u₁|/c ≪ 1, wave vector k_sour is parallel to position vector r_sour-1 from the source to mirror 1 such that it can be written as k_sour = ar_sour-1, where a is a constant at any given time. Similarly, k₁ = br_1–2, k₂ = cr_2–3, where r_1–2 is the position vector from mirror 1 to mirror 2, etc. Using that representation of the wave vectors and substituting (3) into (2) gives, after some algebra given in Hill (2005a),

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where r₃ is the position vector from the source to mirror 3. The reason that k_sour · u₁ does not appear in (4) is that, from (3), u₁ is perpendicular to k_sour and r_sour-1.

Clearly, (4) can be generalized to any number of mirrors. In that case, r₃ is the position vector from the source to the last mirror, and k₃ is the wave vector of the light reflected from the last mirror with frequency ω₃. Also, u₃ = Ω × r₃ is the velocity of the last mirror relative to the source. Therefore, let us write the generalization of (4). For a source and sequence of mirrors constrained to move with a rigid body, the difference between the source frequency and output frequency is given by

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where r_last is the position vector from the source to the last mirror, and u_last is the velocity of that last output mirror relative to the source. According to (5), ω_sour − ω_last contains no effect from mirrors other than the last mirror.

4. Doppler shift resulting from a sequence of mirrors within both the lidar’s seatainer and its scanner

Recall that the hemispheric scanner in Fig. 1 contains two mirrors, the azimuth and elevation mirrors. The scanner rotates about the direction perpendicular to the seatainer’s roof such that the center of rotation of the azimuth mirror does not move relative to the lidar’s optical table; thus, that point can be considered as rigidly fixed relative to the optical table. There is an unobservable small Doppler spread caused by the motion of different positions on the scanner’s two mirrors relative to their center of rotation that is neglected. Henceforth, “center of rotation of the azimuth mirror” is synonymous with “azimuth mirror” and similarly for the elevation mirror.

To determine the frequency of the light output to the atmosphere from the elevation mirror, one must include the effect of the motion of the seatainer and the rotation of the scanner relative to the seatainer. The details are given in Hill (2005a); a synopsis is given here. Recall that motion of the azimuth mirror obeys the rigid-body constraint (3) relative to the source. The frequency shift of the beam reflected from the azimuth mirror as caused by the motion of the lidar’s seatainer is given by (5). At time t let the azimuthal angular velocity of the scanner, relative to the seatainer, be denoted by Ω_scan. The rigid-body constraint is that the velocity of any point within the scanner, relative to the velocity of any point affixed to the seatainer, is

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where r_azim-pt is the position vector from the azimuth mirror to any point in the scanner, and r is the position vector from the chosen seatainer point to that same point in the scanner. If the scanner is not rotating, then Ω_scan = 0, such that (6) becomes the rigid-body constraint (3), as it should because the seatainer and scanner are one rigid body if Ω_scan = 0. If the seatainer is not rotating, then Ω = 0 such that (6) gives a circular motion about the azimuth mirror.

Hereafter, we must distinguish between wave vectors and frequencies observed in the source’s reference frame and those observed in the detector’s reference frame; we do so with superscripts S and D, respectively. Similar to the source’s reference frame, the detector’s reference frame is defined as an inertial reference frame that is moving at the same velocity as the detector at an arbitrary time t. Let the beam reflected from the elevation mirror, as observed from the source’s reference frame, have wave vector k^S_out and frequency ω^S_out; thus, k^S_out and ω^S_out describe the laser beam that exits into the atmosphere. Note that k^S_elev and ω^S_elev would be consistent notation for k^S_out and ω^S_out because subscripts denote spatial points, but we consider the input light at the elevation mirror in the next section and must distinguish it from output light at the elevation mirror. Let r_sour-azim denote the position vector from the source to the azimuth mirror in the scanner and let r_azim-elev be the position vector from the azimuth mirror to the elevation mirror; r_sour-azim + r_azim-elev = r_sour-elev is the position vector from the source to the elevation mirror. Recall that the source’s frequency in the source’s reference frame is denoted ω^S_sour. As shown by Hill (2005a), use of (5) and (6), with the chosen seatainer point being the source and the chosen point in the scanner being the elevation mirror, gives

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where

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From (6), u^S_elev is recognized as the velocity of the elevation mirror relative to the source; (Ω + Ω_scan) is the total angular rate of the scanner. To check (7) we can consider the case when the scanner is not rotating, such that Ω_scan = 0, then ω^S_sour − ω^S_sour = −k^S_out · Ω × r_sour-elev; this is the correct result.

A simple analogy applies. Suppose that the source directly illuminates the azimuth mirror in the scanner and that the reflected wave illuminates the elevation mirror. For that simple case of only one internal mirror, Hill (2005a) showed that use of (1) gives (7)–(8).

5. Doppler shift caused by the return path in the lidar

In the sections above, we followed the beam from its source to its exit into the atmosphere. Light scattered from the atmosphere enters the lidar and traverses a path from the elevation mirror to the detector (see the blue path in Fig. 1). Assume that the motion of the lidar has changed insignificantly during the time that a given lidar pulse has traveled from the elevation mirror to the atmosphere and back to that detector. Then, one can analyze this situation by reversing the direction of the wave vectors in the above analysis and replacing the source’s reference frame with the detector’s reference frame. Recall that the detector’s reference frame is the inertial reference frame that is moving with the detector at time t and is denoted by superscript D. Frequencies and wave vectors in the detector’s reference frame are not the same as in the source’s reference frame. As observed in the detector’s reference frame, let k^D_in and ω^D_in be the wave vector and frequency of light returning from atmospheric scattering which is incident on the elevation mirror. Let ω^D_dete denote the frequency of the returned light at the detector in the detector’s reference frame. Similar to (7)–(8) we have

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where u^D_elev is the velocity of the elevation mirror relative to the detector.

6. The LO frequency at the detector equals the source’s frequency

A light path on the lidar’s optical table from the CW laser source to the detector defines the reference (LO) beam, which is needed to beat with the incoming beam from the atmosphere; that reference beam is the black-to-green path in Fig. 1. There are, in fact, many more reflections on the LO beam path than are shown in Fig. 1. In Hill (2005a) it is shown that

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To be clear, ω^S_sour is the frequency of the source as observed in the source’s inertial reference frame at time t, whereas ω^D_deteLO is the frequency of the LO beam at the detector as observed in the detector’s inertial reference frame at time t. One might anticipate (11) on the basis that there is no relative motion of detector and source parallel to the position vector between source and detector; that fact is easily proven from the rigid-body constraint (3), that is, r · u = r · Ω × r = Ω · r × r = 0. The following three ingredients contribute to (11): (5) and (14), and that (3) applies to all optical elements on the lidar’s optical table.

Although the following is not used in our derivations, it is noted that in any reference frame, ω/c = |k|; therefore, (11) gives |k^S_sour| = |k^D_deteLO|. Of course, the relative directions of k^S_sour and k^D_deteLO depend on the set of reflections on the optical table. On the mini-MOPA’s optical table, those wave vectors are parallel.

7. Relationship of measured Doppler shift to atmospheric Doppler shift

Consider the frequencies, as observed at the detector in the detector’s reference frame, of both the LO light beam and incoming light from the atmosphere. The measured Doppler shift is the difference of those two frequencies. Therefore, the measured Doppler shift is

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The Doppler shift caused by atmospheric scattering is the frequency difference between the incoming and outgoing light, both at the exit side of the elevation mirror; additionally, both must be observed in the same inertial reference frame. Here it is convenient to choose the detector’s reference frame such that

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The subscript “atmos” denotes the atmospheric scatterers within a given lidar pulse volume. For any inertial reference frame denoted by superscript I, in which the velocity of a source is V^I_sour and that of a detector is V^I_dete, the nonrelativistic Doppler shift formula is

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Here, (14) does not require a rigid-body constraint. If I = S, then V^S_sour = 0, V^S_dete ≡ u^S_dete is the velocity of the detector relative to the source, and if the light is the output beam from the elevation mirror, then (14) gives

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Algebraic combination of (7), (9), (11), (12)–(13), and (15) gives

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The rigid-body constraint (3) gives u^S_dete − u^S_elev = −u^D_elev, such that

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where u^D_elev is given in (10). One might anticipate another term in (16); it is zero for the same reason that k_sour · u₁ does not appear in (4).

8. Approximation for (k_out^D − k_in^S)

The quantity (k^D_out − k^S_in) in (16) is the difference of the outgoing and incoming wave vectors of the lidar’s light at the elevation mirror as observed in the source’s and detector’s reference frames, respectively. An approximation for (k^D_out − k^S_in) is required. Alignment of the optics maximizes the received power, and we neglect the change of orientation of the elevation mirror during the time the light is in the atmosphere; therefore, k^D_out and −k^S_in are approximately aligned with the unit vector p that defines the outbound direction of the lidar’s beam. Then,

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The magnitudes of the wave vectors are

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The difference between ω^S_out and ω^D_in generates terms in (16) that are second order in the differences between the velocities of the scanner, detector, and source (see Hill 2005a). Therefore, let us simplify by approximating those frequencies by ω_lidar [e.g., ω_lidar = (ω^S_out + ω^D_in)/2], such that

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then,

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9. Radial component of atmospheric velocity

Let E (mnemonic for earth-fixed GPS coordinate system) denote the inertial reference frame that is moving with the velocity of the earth-fixed GPS coordinates at any chosen position at time t. In this section that position is chosen to be that of the lidar’s detector. In subsequent sections we need not specify the chosen position. An alternative to use of reference frame D in (13) is to use E, that is,

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Recall that subscripts “out” and “in” refer to the outbound and inbound lidar beams within the atmosphere, respectively. A given velocity observed in the GPS coordinates is denoted by a superscript E, for example, V^E. We can rewrite the Doppler shift Eq. (14) using different inertial reference frames and positions; consider, for example,

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where A denotes the inertial reference frame moving with the atmospheric scatterers that are within a specific lidar pulse volume at time t. The lidar’s light is scattered in all directions; consider only the light that is scattered back toward the elevation mirror. In reference frame A, that light is scattered without a change of frequency or wavelength, but the wave vector is reversed in direction; that is,

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Similar to (19) we can write

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Add (19) and (21) and, from (20), substitute ω^A_out = ω^A_in to obtain

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We can use the approximation (17) for both wave–vector differences in (22) and use (18) to obtain

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Despite the several reference frames in motion relative to one another, (23) is the usual formula for the velocity of atmospheric scatterers when a lidar or radar is stationary relative to the earth-fixed GPS reference frame (for which V^E_dete = 0). It is important that the earth-fixed GPS frame velocity of the scatterers relative to the earth-fixed GPS frame velocity of the lidar’s detector appears in (23) for the case V^E_dete ≠ 0.

We must eliminate Δω_atmos using (23) and (16) because Δω_atmos is not a measured quantity. Substitute (17) into (16). The velocity of the elevation mirror relative to the detector, namely, u^D_elev, equals V^E_elev − V^E_dete such that (16) becomes

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Substitute (24) into (23) to eliminate Δω_atmos to obtain

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The radial component of atmospheric velocity is p · V^E_atmos; recall that p is the unit vector in the direction of the outbound lidar beam. To use (25) to determine p · V^E_atmos, we must obtain p · V^E_elev using data from a ship- motion detection system.

10. Velocity from the motion detection systems

A motion detection system (MDS) is needed in combination with the lidar’s data to determine the atmospheric velocity in the GPS coordinate system. The lidar’s motion detection system JAVAD consists of four GPS antennas, which constitute the primary motion measurement, and a six-axis accelerometer. During the RICO experiment we recorded motion data from the ship’s POS MV. The POS MV has a strapped-down inertial navigation accelerometer system that provides the primary motion measurement with GPS data for long-term corrections. The description of the data recorded by the POS MV system is given in Corcoran and Pronk (2003). Both the lidar’s and ship’s coordinate systems are rotating relative to the GPS coordinate system by angular rate denoted by Ω, which is measured by both motion detection systems. The lidar’s motion compensation system reported the velocity relative to GPS coordinates at the location of its master GPS antenna shown in Fig. 2. The POS MV reported velocity at a point 22 m aft of the scanner; the precise POS MV location is given in Hill (2005b). Whether we use the lidar’s motion compensation system or the ship’s POS MV data, we denote the velocity as V^E_MDS, where the subscript MDS denotes the point at which either system reports velocity. Consider a position vector r that points from the MDS point to any other location on the ship. Let V^E_r be the velocity of that location on the ship in the earth-fixed GPS coordinate system. According to rigid-body kinematics,

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The vector’s Ω and r are most conveniently given in the coordinate system of either the lidar or ship. Hence, the notation Ω × r implies performing the cross product in the lidar’s or POS MV’s coordinate system, and then transforming to the earth-fixed GPS coordinate system to obtain (Ω × r)^E; the Euler rotation matrix is used for that transformation. The Euler angles are obtained by temporal integration of Ω.

The velocity of the elevation mirror in the GPS coordinates is denoted by V^E_elev. Denote the position vector from the MDS point to the scanner’s azimuth mirror by r_MDS-azim. Application of (26) gives

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Recall that r_azim-elev is the position vector from the azimuth mirror to the elevation mirror.

11. Correction of measured Doppler shift to obtain the radial component of atmospheric velocity

Substitution of (27) into (25) gives

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All quantities on the right-hand side of (28) are measured quantities. If Ω_scan = 0 then (28) simplifies to

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where

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is the position vector from the MDS point to the elevation mirror. When the mini-MOPA’s scanner rotates in azimuth, |Ω_scan × r_azim-elev| = (0.052 s⁻¹) (0.43 m) sin (π/2) = 0.022 m s⁻¹, such that (29) is an adequate approximation provided that one uses the fact that r_MDS-elev is a function of time because of the rotation of the elevation mirror around the azimuth mirror (in the ship and lidar reference frames). When the lidar is in zenith stare mode, the scanner is stationary relative to the seatainer such that Ω_scan = 0 and (29) applies, and r_MDS-elev is constant.

12. Motion correction of RICO lidar data

The motion correction obtained using (29) is illustrated using data from RICO. The correction is best illustrated for the zenith stare mode because spatial variation of horizontal wind components observed in scan mode makes the motion contamination less obvious. The hemispherical scanner maintains the pointing of the lidar beam to within 0.5° despite the motion of the ship. During the zenith stare mode, that maintained pointing is to the zenith such that the lidar measures the vertical component of velocity at all heights that have adequate signal-to-noise ratio (SNR). The vertical component of V^E_MDS + (Ω × r_MDS-elev)^E is used in the zenith stare mode for calculation of the correction in (29). Denote the unit vector pointing toward zenith by z, that is, for zenith stare p = z in (28)–(29). Recall that the lidar and ship coordinate systems have a positive keelward axis. Locally, the earth-fixed GPS coordinate system is eastward, northward, and upward, in that order. Our figures show the vertical velocity component in the earth-fixed GPS coordinate system such that upward velocity is positive.

The NOAA/ESRL mini-MOPA lidar was deployed during the RICO experiment. Figure 3 (upper panel) shows the SNR for a 15-min time segment during RICO. A cloud base is evident from the patch of enhanced SNR at 2-km altitude. Also in Fig. 3 are the vertical components of velocities in the GPS coordinates as calculated from the lidars’ motion compensation system data during 2 min within the 15-min segment. The vertical components in Fig. 3 are z · V^E_MDS, z · (Ω × r_MDS-elev)^E and z · [V^E_MDS + (Ω × r_MDS-elev)^E]. Figure 4 shows uncorrected and corrected vertical velocity components obtained by application of the lidar’s motion detection system data in (29). The ship’s motion produces vertical stripes in the uncorrected data that are effectively removed by the correction.

Figure 5 (upper panel) shows SNR for 18 min during RICO when more clouds and rain were present relative to Fig. 3. Figure 5 (lower panel) shows vertical velocity components during the first 2 min of data in the upper panel as obtained from both the lidar’s motion compensation system and the ship’s POS MV. The JAVAD master GPS antenna and the scanner are on opposite sides of the ship’s roll axis; therefore, Ω × r_MDS-elev from the lidar’s motion detection system is caused more by variation of the ship’s roll than by variation of pitch or heading. In contrast, the scanner is nearly straight forward (in the ship’s coordinate system) from the point where the POS MV gives its V^E_MDS; hence, the ship’s pitch mostly determines Ω × r_MDS-elev from the POS MV data. Despite those differences, Fig. 5 shows that the calculated vertical velocity component of the elevation mirror z · [V^E_MDS + (Ω × r_MDS-elev)^E] is nearly the same for both the lidar’s motion detection and POS MV systems. Figure 6 shows the uncorrected and corrected vertical velocity components obtained by use of the lidar’s motion detection system data in (29). Data in Fig. 6 correspond to the same 18 min that are shown in Fig. 5. Note the strong downward velocity caused by rain. The ship’s motion is effectively removed by the motion correction.

The data in Fig. 4 can be used to estimate an upper bound for velocity variance caused by uncompensated ship motion as follows. To detect uncompensated ship motion, the ideal case is to measure velocity in an atmosphere that is in uniform motion; then, the measured velocity variation would be caused by ship motion and instrument noise. To emulate that ideal case using the data in Fig. 4, we create a time series of the height average of the vertical velocities shown in both panels of Fig. 4 from 330- to 1350-m heights. That height range is chosen because Fig. 3 shows good signal-to-noise ratio below 1350 m. Uncompensated ship motion for a single vertical velocity profile affects the entire column of heights equally and creates the striped pattern that dominates the uncompensated vertical velocities in the upper panel in Fig. 4. The motion-compensated vertical velocities in the lower panel in Fig. 4 are dominated by the atmospheric fluctuations that change in both height and time. The height average reduces the contribution of the atmospheric vertical velocity variation present within each column of height without modifying the contribution of the uncompensated ship motion. The resulting height-averaged time series contains low-spatial-frequency atmospheric variations and high-frequency uncorrelated random noise. To minimize these variations, we apply a bandpass filter to the time series. For the data of Fig. 4, ship motion is sharply reduced below 0.065 Hz and it reaches the noise level at about 0.29 Hz. Therefore, the corner frequencies for the bandpass filter are chosen to be 0.065– 0.29 Hz.

The variance of the resultant height-averaged and bandpassed signal is called the “total” variance. Noise variance is calculated from the height-averaged time series at frequencies above 0.3 Hz and that noise variance is scaled to represent the noise within the bandpass filter. The “signal” variance is the total variance minus the noise variance. The square root of those three variances is the root-mean-squares (rms) of the total, noise, and signal. Table 1 gives the total rms of the height-averaged, bandpass-filtered time series from the data in Fig. 4 and the rms noise and signal. Motion-compensated data in the lower panel of Fig. 4 produced the “compensated” rms (given in Table 1), whereas data in the upper panel of Fig. 4 produced the “uncompensated” rms. We can see from Table 1 that noise has a small effect and that the height-averaged, bandpass-filtered signal rms is reduced from its motion-uncompensated value by a factor of 6.4 to its motion-compensated value of 0.075 m s⁻¹.

Thus, 0.075 m s⁻¹ is an upper bound to the rms error caused by uncompensated ship motion or by other error sources. It is an upper bound because we cannot estimate the atmospheric velocity variance that exists despite the height average and bandpass filtering. Our upper bound of 0.075 m s⁻¹ is above the uncorrelated noise of 0.000 46 m s⁻¹ given in Table 1.

To determine how important the 0.075 m s⁻¹ upper-bound error is, we compare it to the atmospheric vertical velocity measurements in the lower panel of Fig. 4 without any height averaging or bandpass filtering. Table 2 gives the rms values for motion-uncompensated and motion-compensated data from the height of 330 m. Motion-compensated data at 330 m in the lower panel of Fig. 4 produced the compensated rms, whereas data at 330 m in the upper panel of Fig. 4 produced the uncompensated rms. The compensated signal variance in Table 2 is our best estimate of atmospheric variability. The upper bound of 0.075 m s⁻¹ for the rms error caused by uncompensated ship motion or by other error sources is 5.4 times smaller than this motion-compensated signal rms in Table 2. In this case we have removed ship motion accounting for 0.49 m s⁻¹ of rms vertical velocity variability. Of course, the rms of horizontal velocity components exceeds that of the vertical component, and the average wind velocity typically greatly exceeds the rms values.

13. Summary

The main motivation of this report is presentation of the derivation for the motion correction of seagoing NOAA/ESRL Doppler lidars. The lidars are spatially extended systems; therefore, our derivation takes into account the facts that the lidar’s optical elements move with differing velocities and that the velocity reported by the motion detection system differs from the velocity of the optical elements. The main result is the correction formula from that derivation. The main application is to the motion correction of Doppler lidar data of the NOAA/ESRL lidars. We now summarize that main motivation and that main result and the example application. That is, we summarize the body of this paper, specifically sections 2–12. Specific reference to equation numbers and sections is intended to aid the reader, but it is not necessary for the reader to refer back to those items in the body of this paper.

We give a derivation of the correction of Doppler measurements of velocity for platform motion. The derivation is general enough to apply to radar and lidar and any moving platform. We exemplify our derivation by analyzing the NOAA/ESRL mini-MOPA Doppler lidar. The mini-MOPA’s seatainer and scanner are two rigid bodies that rotate relative to one another. For reflections from an arbitrary number of mirrors (or beam splitters, etc.) constrained to a rigid body, we show that only the velocity of the last mirror relative to the source determines the frequency difference between the source and output beams [see (5)]. In particular, it is the component of velocity parallel to the output wave vector that appears in (5). That statement remains true for beams reflected within two rigid bodies that can rotate relative to one another, as shown in (7)–(8). The analogous result is true for the scattered light returning to the detector, as shown in (9)–(10). Those results can be generalized to any number of rigid bodies attached to one another and free to rotate relative to each other. For a source and detector attached to the same rigid body, we show that the LO frequency at the detector in the detector’s reference frame equals the source’s frequency in the source’s reference frame [see (11)]. Combining the aforementioned results with the nonrelativistic Doppler shift Eq. (14) gives the correction [(16)] of the measured Doppler frequency shift to obtain the Doppler shift caused by atmospheric scattering; that correction depends on the velocity relative to the detector of the mirror that outputs the beam to the atmosphere and on a wave–vector difference; the latter is approximated in (17). Four different inertial reference frames are used in the derivations; therefore, we derive the relationship (23) of Doppler shift caused by atmospheric scattering to the velocity of scatterers; the important aspect of (23) is that the relevant velocity is that of the scatterers relative to the lidar detector’s reference frame. All of the foregoing is used to express the radial component of the velocity of atmospheric scatterers in the earth-fixed GPS coordinate system in terms of the measured Doppler shift and the corresponding radial component of the instrument’s output mirror [see (25)]. In section 10 we discuss the use of a motion detection system to obtain both the translation velocity of the output mirror as well as its motion caused by ship rotation. In section 11, the desired motion correction formula is given; namely, the radial component of the velocity of atmospheric scatterers in the earth-fixed GPS coordinate system is given in terms of the measured Doppler shift and data from the motion detection system. Our analysis is for a monostatic system. A bistatic system would be more complicated; for example, (17) would depend on two directions, and the mirror positions in (10) would not be the same as in (8). Throughout the derivations we rely on the following two facts: 1) the speed of light greatly exceeds all other velocities, and 2) the instrument’s translatory motion and change of orientation are negligible during the time required for light to travel from the source to the scatterers and back to the detector. As such, our analysis might not apply to acoustic Doppler systems.

In Figs. 3 –6 we show two examples of successful motion correction obtained from the RICO experiment. In the lower panel of Fig. 6 we show excellent agreement of velocities from the ship’s POS MV and the lidar’s motion compensation system. That agreement shows that it is possible to use a ship’s navigation system for the purpose of motion correction.

14. Comparison with other work and significance

Here, we must compare our derivation with that of Edson et al. (1998). They derive the motion correction for an in situ sensor mounted on a moving platform. Specifically, they consider a sonic anemometer aboard a ship. Denote the atmosphere’s velocity vector measured by the anemometer as V^E_obs. Superscript E (mnemonic for earth-fixed GPS coordinate system) denotes that the vector’s components are expressed in earth-fixed GPS coordinates at the moment of the measurement t. The velocity measured by a sonic anemometer gives the components of velocity parallel to the acoustic paths of the sonic anemometer, whose paths are fixed relative to the ship’s coordinate system. Thus, the notation V^E_obs implies that at each time t a rotation is performed to transform the velocity components from the ship’s coordinate system to the GPS coordinate system. Denote the wind’s velocity vector that would be observed by an anemometer that was not moving relative to the earth-fixed GPS coordinate system and was at the same location as the shipborne sonic anemometer at the time t as V^E_wind. In our (26), let the anemometer be at the location on the ship denoted by the tip of the vector r. To obtain V^E_wind we add our (26) to V^E_obs to obtain

View Expanded

This is the same as Eq. (10) of Edson et al. (1998). We obtain this result more simply than Edson et al. (1998) did because we did not begin with the ship’s center-of-mass coordinate system, which Edson et al. (1998) did in their Eq. (4). They point out that the location of the ship’s center of mass is unknown. Also, its location varies with the ship’s ballast trim. It is conceptually not useful because the center of mass is not in uniform motion because of ocean forcings of the ship. If we use the center-of-mass coordinate system to express the velocities of two points on the same rigid body and algebraically eliminate the velocity of the center of mass between those two equations, then we obtain (26).

Here, we must further distinguish our derivation from that of Edson et al. (1998) by the fact that they did not consider a Doppler velocity measurement. The lidar is a spatially extended system such that ship motion causes a change of the frequency of the lidar light beam at each reflection of the beam within the system. We followed the lidar’s light beam from the source to the azimuth mirror [on which path there are about 23 reflections; cf. Fig. 1 with Hill (2005a)]. We then followed the beam to the elevation mirror (two reflections), to the atmospheric scattering (one reflection), back to the azimuth mirror (two reflections), and from there to the signal detector (about seven reflections), where the beam was mixed with the LO beam (the LO beam underwent about six reflections from the source to the detector). Between each pair of reflections the beams have distinct frequencies, and hence distinct wavelengths as well. Our final result in (28) is that the motion correction depends on the velocity of the seatainer and on the motion of the hemispheric scanner relative to the seatainer.

We must state that the motion compensation has been performed for entire datasets for all scan directions for every shipborne experiment since the capability was developed. We stated herein that the motion correction “is illustrated using data from RICO.” Only a short time interval and only for zenith stare mode is lidar motion correction shown in this paper. Doppler lidar has had enormous impact on boundary layer meteorology (see Hardesty 2002; Banta et al. 2003); its extension to the liquid portion of the earth’s surface cannot be underestimated. Motion correction is a prerequisite for such extension.