a. Motion-correction equation

We define three right-handed coordinate frames. The earth-reference frame (subscript e) is defined with x axis eastward, y axis northward, and z axis up. The origin of this frame translates with the ship. The platform frame (subscript p) is defined by the instantaneous location and orientation of the three orthogonal motion sensor coordinate axes. Similarly, the anemometer frame (subscript a) is defined by the instantaneous location and orientation of the three orthogonal sonic anemometer coordinate axes. The platform and anemometer coordinate frames translate, pitch, roll, yaw, and heave with the platform, with x axis (nominally) toward the bow and y axis toward port (Fig. 1). For a right-handed system, positive roll refers to portside up, positive pitch refers to bow down, and positive yaw refers to counterclockwise rotation.

A vector a in coordinate frame α is denoted a_α. This vector is transformed to coordinate frame β by multiplication with the rotation matrix 𝗧_βα; that is, a_β = 𝗧_βαa_α. A coordinate transformation can be defined by three Euler angles (ϕ, θ, ψ) as 𝗧_βα = [ψ_βα][θ_βα][ϕ_βα], where the first rotation is about the x axis (roll ϕ), the second rotation is about the intermediate y axis (pitch θ), and the final rotation is about the new intermediate z axis (yaw, or heading, ψ), and the square brackets denote rotation matrices. The individual rotations are expressed in matrix form as

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The true wind vector, u = (u_east, u_north, w), is the sum of the apparent wind vector measured by the anemometer and expressed in the anemometer frame u_a and the velocity of the anemometer expressed in the platform frame, v_p + v_ship, where v_p is the velocity of the anemometer due to ship motions (pitch, roll, yaw, and heave), and v_ship is the velocity of the ship recorded by the ship’s navigation system or a GPS receiver. Expressing all terms in the earth coordinate frame,

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where the transformation 𝗧_ea rotates the wind vector measured in the anemometer’s frame to the earth frame, and 𝗧_ep rotates the velocity of the anemometer measured in the platform frame to the earth frame. Note that if the anemometer and platform (i.e., motion sensor) coordinate axes are collinear, 𝗧_ea = 𝗧_ep.

Following E98, we focus on strapped-down motion sensors that measure three linear accelerations and three angle rates. It is convenient to separate the anemometer velocity (v_p) into linear and angular motion terms, v_p = ∫_p dt + Ω_p × r_p, where _p = (_p, ÿ_p, z̈_p) is linear acceleration, Ω_p = (ω_x, ω_y, ω_z) is the angular velocity vector, and r_p = (x_p, y_p, z_p) is the position vector from the motion sensor to the anemometer. We assume that r_p is known and constant, and Eq. (2) becomes

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Equation (3) is analogous to E98, except that for their mounting arrangement the anemometer and motion sensor axes were collinear so that 𝗧_ea = 𝗧_ep.

b. Measurements from R/P FLIP and R/V Wecoma

We focus on data collected using the research platform R/P FLIP, a 120-m-long spar buoy designed for stability in the vertical orientation (Fig. 1). FLIP was moored 50 km off the coast of California (water depth 1500 m) for 12 days during the marine boundary layer (MBL) experiment in spring 1995 (27 April–9 May, days of year 117–128.5). A three-point mooring maintained a keel heading of 350° ± 15°, and FLIP remained in a fixed location such that v_ship = 0 in Eq. (3). A 16-m-long vertical mast (triangular cross section with 23-cm side length) was fixed to the end of FLIP’s portside boom by a hinge mechanism that allowed the mast to pivot horizontal for installation of sensors (Fig. 1). The nominal x axis of the motion sensor and anemometer coordinate frames pointed to FLIP’s keel, and the nominal y axis outward along the portside boom.

The turbulent wind velocity plus contributions due to motion, u_a + v_p + v_ship, was measured using four three-dimensional ultrasonic (sonic) anemometers (Model 1012, Gill Instruments, Ltd., Great Britain). Three sonics were mounted along the vertical mast at the end of FLIP’s portside boom, at 8.7, 13.8, and 18.1 m above the mean ocean surface. Each sonic was mounted to a 3.8-cm-diameter aluminum pipe that extended 2 m from the vertical mast in the portward direction. A fourth sonic was hung upside-down from the port boom using a 5.1-cm-diameter aluminum pipe, approximately 5 m inward from the mast and 3.5 m above the mean ocean surface. The alignment of the three sonic anemometers fixed to the mast was sighted by eye, while the mast lay flat on the portside boom. The mast was then pivoted to the vertical position, as shown in Fig. 1.

Motion was measured using a 6-degrees-of-freedom inertial sensor (Systron-Donner MotionPak) mounted at the end of the portside boom at the hinge point of the mast. The motion sensor consisted of three orthogonal linear accelerometers (to measure _p, resolution <10 μg) and three orthogonal angle rate sensors (to measure Ω_p, resolution <0.004° s⁻¹), and was the same unit used by E98. The locations of the anemometers mounted to the vertical mast relative to the motion sensor are listed in Table 2. FLIP’s compass heading was recorded by the ship gyroscope. Wind and motion data were recorded at 50 Hz and subsequently resampled to 5 Hz for processing.

To demonstrate the effectiveness of FLIP for making wind and wind stress measurements, we compared the FLIP data with in situ data collected from the R/V Wecoma, a 56-m-long conventional research vessel that operated near FLIP during the 1995 MBL campaign. The wind vector and ship motion were measured by sensors at Wecoma’s bow, approximately 10 m above the mean ocean surface. A critical difference between the FLIP and Wecoma datasets was that, for Wecoma, the relative mounting between the motion sensor and anemometer was fixed using a machined mounting bracket. Because of the fixed alignment of the anemometer and motion sensor, there was no correction required for misalignment between the two sensors for the Wecoma data [i.e., 𝗧_ea = 𝗧_ep in Eq. (3)]. Therefore, the comparison between motion corrections for the FLIP and Wecoma datasets demonstrates the sensitivity to different amounts of motion experienced by different platforms measuring in situ during the same wind and wave conditions. The comparison of the importance of accounting for sensor misalignment will be presented by comparing motion corrections with and without accounting for the mounting offsets using data collected from the four sensors on FLIP.

c. The transformation matrix

The procedure for calculating the transformation 𝗧_ep was detailed by E98. To summarize, the time-dependent Euler angles were calculated in two ways: the first is based on integration of the angle rate sensors and the second on the ratio of measured horizontal acceleration components to the gravity acceleration, g, ϕ ≈ tan⁻¹(ÿ_p/g) and θ ≈ tan⁻¹(−_p/g). The two estimates were combined using complimentary filtering, where high-pass-filtered rate-sensor-based angle estimates were combined with low-pass-filtered estimates from the accelerometers. We used a fourth-order Butterworth filter to define the complementary filter, with a cutoff period of 50 s, and the filter was applied forward and backward on each motion channel to eliminate phase shifts (e.g., Oppenheim and Schafer 1989). We found that the use of a higher-order filter than the first-order filter used in E98 (and that corresponds to the definition of the complimentary filter) resulted in less “leakage” between the angle-rate-sensor-based and accelerometer-based Euler angle estimates. The low-frequency yaw was obtained from a gyro on FLIP and from the GPS receiver on R/V Wecoma, and 𝗧_ep was calculated as the matrix product of the individual rotations, 𝗧_ep = [ψ_ep][θ_ep][ϕ_ep].

The platform linear velocity was calculated by integrating the linear accelerations in the earth frame, ∫𝗧_epp dt. The numerical integration amplified low-frequency drift of the accelerometers, and high-pass filtering was necessary after integrating (also observed by E98). The filter cutoff period was selected empirically by plotting velocity variances and covariances versus a range of values for the filter cutoff, and by choosing values at which statistics became insensitive to further increases in cutoff period (Fig. 2). The filter cutoff period for the vertical acceleration was 40 s, and a longer period (80 s) was used for the horizontal accelerations to account for low-frequency rotation of FLIP about its vertical axis. We note that E98 used the same filter cutoff for the three acceleration components. To verify our choice of accelerometer filter cutoff periods, the calculated platform velocity was integrated to obtain the platform displacement at the end of the portside boom (Fig. 2). We compared the calculated displacement with that expected for pitch and roll of FLIP about its center of mass (approximately 35 m below the ocean surface). During high wave conditions with relatively large motion, the range of vertical displacements was less than 1 m. For filter cutoff periods greater than 40 s, the calculated range of vertical displacement at the end of the portside boom increased linearly, consistent with the amplification of low-frequency accelerometer drift (Fig. 2).

d. Instrument-mounting offsets

The true mean vertical wind component w in the surface layer was expected to be close to zero; however, at all levels the measured velocity along the z axis of the sonic anemometer was nonzero and increased in magnitude with horizontal wind speed (Fig. 3, gray x’s). Two mechanisms could have produced this result: 1) the anemometers were tilted with respect to a horizontal plane such that horizontal wind components affected apparent wind along the anemometer z axis; and 2) the airflow was distorted by FLIP, the portside boom, the instrument mast, and/or the instrument mounts (Yelland et al. 2002). The horizontal distance between the mast and FLIP’s hull was 20 m, more than 2.5 times FLIP’s widest horizontal dimension. This relatively large separation suggests that flow distortion due to the portside boom would have a greater impact on the airflow at the location of the mast (Mollo-Christensen 1979). However, the measured z component of the wind was negative at all levels, whereas flow distortion about the portside boom would be expected to deflect the flow in opposite directions for levels above, versus below, the boom. Based on these observations and the three-point mooring that maintained optimal exposure of the instrument mast, we conclude that the effect of anemometer tilting was likely the primary cause of nonzero wind components along the anemometer z axes.

The first attempt at applying motion correction to the FLIP dataset resulted in corrected mean vertical winds of order 10–100 cm s⁻¹ during high wind periods (Fig. 3, solid dots). This suggested mounting offsets of each of the anemometers with respect to the motion sensor. One source of alignment offset between the sonic anemometers and the motion sensor was caused by nonorthogonality between the top face of the port-side boom (on which the motion sensor was mounted) and the hinge mechanism to which the vertical mast was attached. In addition, at one point during the experiment waves struck the bottom of the mast and caused a relative shift in the mounting offsets between the motion sensor and anemometers. Finally, the mounting braces used to attach the anemometers to the mast were modified during setup to avoid interference with guy wires supporting the port boom during rotation of the mast from horizontal to vertical. This modification added uncertainty in the mounting orientation of the anemometers relative to the mast and motion sensor.

The complementary filter described in section 2c accounted for tilting of the motion sensor with respect to earth but did not account for mounting offsets of the anemometers with respect to the motion sensor. We define a transformation 𝗠_pa as the relative mounting offset between an anemometer and the motion sensor. Compared to the transformation 𝗧 that is updated at the sampling rate of the turbulence and motion sensors (5 Hz for this dataset), the transformation 𝗠 is fixed and only changed if the relative offset between the anemometer and motion sensors changed, for example, after the mast was struck by waves. The transformation 𝗧_ea in Eq. (3) was rewritten in terms of 𝗧_ep according to 𝗧_ea = 𝗧_ep𝗠_pa,

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In E98, the motion sensor and anemometer were mounted together so their coordinate axes were coaligned and 𝗠_pa = 𝗜, where 𝗜 is the identity matrix. For the small offset angles we measured (less than 7°, Table 2), we found that the matrix multiplication 𝗧_ea = 𝗧_ep𝗠_pa in Eq. (4) was closely approximated by simply adding the offset angles to the calculated Euler angles, that is, 𝗧_ep𝗠_pa = [ψ_ep][θ_ep][ϕ_ep]𝗠_pa ≈ [ψ_ep][θ_ep + (θ_ep − θ_ea)][ϕ_ep + (ϕ_ep − ϕ_ea)].

The mounting offset angles between anemometers and the motion sensor were estimated using measured acceleration and wind signals. The transformation 𝗠_pa was separated into two transformations that were each estimated from the measured data, 𝗠_pa = 𝗠⁻¹_ep𝗠_ea, where 𝗠_ep is a transformation from the motion sensor to the earth frame, and 𝗠_ea is a transformation from the anemometer frame to the earth frame. We estimated 𝗠⁻¹_ep by assuming that the long-term average horizontal acceleration was zero. With this assumption, nonzero measured mean horizontal accelerations were due to tilting of the motion sensor relative to vertical (i.e., gravity). We estimated 𝗠_ea by assuming that the long-term mean vertical wind velocity over the ocean was zero (in the absence of flow distortion), and nonzero measured mean vertical wind was due to tilting of the anemometer relative to the horizontal plane. The calculations are similar to the “planar fit” method presented by Wilczak et al. (2001). Equivalent roll (ϕ_ea) and pitch (θ_ea) offset angles of each anemometer with respect to earth were calculated using a least squares regression on (long) records of measured 30-min mean velocity components (, , ). The regression resulted in pitch and roll angles that minimized the mean vertical wind velocity over long intervals (days). The transformation 𝗠_ea was calculated as

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A similar approach was used to calculate 𝗠_ep using long records of the 30-min-averaged measured acceleration components.

The calculated motion sensor and anemometer mounting offset angles relative to earth are shown in Table 2. The offset angles for the anemometers mounted to the mast were less than ±4°. It was necessary to calculate the offset angles for two distinct periods, one before and the other after the mast was struck by high waves (Table 2).

e. Summary of correction algorithm

For the FLIP dataset, the mounting offset angles for the motion sensor (ϕ_ep, θ_ep) and for each anemometer (ϕ_ea, θ_ea) were calculated in a preprocessing step using 30-min-averaged measured wind speed and acceleration over the entire dataset. A single estimate of the offset angles was calculated by linear regression using all of the data during a period when the mean offset angles did not change. A step-change in the offset angles was clear after the mast was struck by waves. The offset angles calculated separately for the period before and after waves struck the mast are shown in Table 2. The offset angles of the 3.5-m sonic fixed to the pole were dependent on wind speed due to wind drag on the pole, and its offset angles were updated more frequently [thus, the nonlinear relationship between u and w in Fig. 3 (bottom)]. For this anemometer, a continuous time series of offset angles was generated using a sliding window: for each time period, forty-eight 30-min intervals nearest to the interval were used to calculate the offset angles. The resulting time series was smoothed with a running mean filter. For this sensor, the average of the time series before and after the mast was struck by waves is shown in Table 2.

The offset angles of the motion sensor and each anemometer were used to calculate the fixed transformation matrices 𝗠_ep, 𝗠_ea, and 𝗠_pa according to the procedure in section 2d. For each flux interval, the “fast” transformation matrix 𝗧_ep was calculated as outlined in section 2c and described in detail in E98. The remaining terms in Eq. (4) were measured. For the R/V Wecoma data, the anemometer and motion sensor were rigidly mounted with coaligned coordinate axes (as in E98), such that the preprocessing step was unnecessary.

After motion correction and before calculating turbulence statistics, with or without mounting offset angles, the wind vector was rotated into the 30-min mean wind direction such that the mean υ and w components were zero (Donelan 1990). This rotation into the streamline (or “natural”) coordinate does not correct for motion effects but does affect turbulence statistics. We evaluate the effect of the mounting angle offsets on the turbulence statistics by comparing results calculated in the natural coordinate frame using 𝗠_pa = 𝗠 ^{− 1}_ep𝗠_ea and results calculated using 𝗠_pa = 𝗜.

a. Mean vertical wind

Our first attempt at motion correction applied Eq. (4) and assumed that 𝗠_pa = 𝗜; that is, it accounted only for the orientation of the motion sensor with respect to earth, not for mounting offsets between the anemometer and motion sensor coordinate frames. The “corrected” vertical wind without accounting for mounting offsets also showed significant nonzero w at all levels (Fig. 3, solid dots). We then calculated the mean mounting offsets (𝗠_pa) according to the procedure in section 2e and recalculated the motion correction according to Eq. (4). With the mounting offsets taken into account, the mean vertical wind was effectively removed by the motion correction (Fig. 3, open circles). Residual mean vertical velocities were generally less than 10 cm s⁻¹ at all levels. We note that 𝗠_pa minimized w over long periods (days) but did not require that the calculated vertical velocity for a 30-min interval be zero. As mentioned previously, an additional rotation into the natural coordinate frame for the flux record was performed before calculating turbulence statistics, such that w = = 0 (Donelan 1990).

b. Effect of motion on spectra

Measured and motion-corrected velocity spectra and cospectra are shown for the four levels in Fig. 4 for a 60-min interval with moderate wind speed (7 m s⁻¹ at 8.7-m height). Spectral peaks at the surface wave period (5–15 s) were found in the measured u- and υ-component spectra (Fig. 4, solid curve in rows 1 and 2). The u and υ spectral peaks were reduced by the motion correction at all levels, particularly 8.7, 13.8, and 18.1 m (solid curve with circles). The peaks at the lowest level were only partially removed by motion correction, suggesting real wind fluctuations near the sea surface correlated with the surface waves (e.g., Hristov et al. 2003). The w-component spectra support the presence of wave-correlated fluctuations, as the corrected spectral peak at the surface wave period was largest at the lowest level and decreased with distance from the surface (Fig. 4, row 3). Compared to the u and υ components, the motion correction had little effect on the vertical velocity spectrum, presumably because FLIP’s stability minimized platform vertical velocity.

The uncorrected along-wind stress cospectra were negative at all four levels corresponding to downward momentum transfer, except for a sharp positive peak at the surface wave frequency (Fig. 4, row 4). The motion correction removed most of these peaks. For this interval, the motion correction had little effect at high and low ends of the cospectrum. The uncorrected crosswind stress cospectra also showed peaks at the surface wave frequency (Fig. 4, row 5). The motion correction removed most of these peaks, and the corrected crosswind cospectra were close to zero.

The mean tilt angle correction had negligible effect on velocity spectra and cospectra at the surface wave and higher frequencies during the interval shown in Fig. 4 (dashed curves). The mean tilt correction had a small effect on the low-frequency vertical power spectrum and a larger effect on the low-frequency υw cospectra. The net effect of the mean sonic tilt angle correction on velocity variances and covariances is discussed below.

c. Wind velocity variances and covariances

The effect of motion corrections on velocity variances is expressed as σ²_meas/σ²_corr, where σ²_meas is the measured variance and σ²_corr is the corrected variance. Table 3 summarizes the corrections for 6 days of measurements with the most favorable relative wind directions. The 10-m wind speed varied between 1 and 13 m s⁻¹. The average error magnitude was 5% for σ²_u, 15% for σ²_υ, and 3% for σ²_w. The insensitivity of σ²_w to motion was presumably due to the stability of FLIP. The signed average error was less for the individual levels (Table 3). The correction to σ²_υ showed a systematic dependence on measurement height, while σ²_u and σ²_w did not. The measured σ²_υ was overestimated by 18% at the 18-m level and decreased to 10% at the 3.5-m level. The effect of mean sensor tilt had a negligible (<1%) effect on each of the velocity component variances.

The velocity variance correction showed a different dependence on the 10-m wind speed for each of the wind components (Fig. 5). The shapes of the curves in Fig. 5 were similar when plotted against FLIP’s motion, calculated as the difference between the maximum and minimum Euler angles in a 30-min flux record (Δϕ, Δθ, or Δψ). Regressions between wind speed and motion were as follows: Δϕ = 0.02 U²₁₀ − 0.02 U₁₀ + 1, r ² = 0.39; Δθ = 0.07 U²₁₀ − 0.6 U₁₀ + 2.3, r ² = 0.5; and Δψ = 0.07 U²₁₀ + 0.2 U₁₀ + 0.7, r ² = 0.89. The measured σ²_u was overestimated by approximately 15% for wind speeds less than 6 m s⁻¹. Above 6 m s⁻¹, σ²_u was less than 5% and was not wind speed or motion dependent (Fig. 5, top). In contrast, the measured error in σ²_υ was smallest for wind speeds between 6 and 8 m s⁻¹ and was larger for higher and lower wind speeds. The σ²_υ correction showed a systematic height dependence for wind speeds above 8 m s⁻¹, with larger errors at the higher measurement levels (Fig. 5, middle). The motion correction to σ²_w was small across all wind speed values and all measurement heights (Fig. 5, bottom).

Combining the four measurement levels, the average magnitude of wind stress error for 270 30-min intervals (10-m wind speed range 2–14 m s⁻¹) was 15% ± 1% (Table 3). Without accounting for sensor tilt offsets, the error average magnitude was 5%, indicating that one-third of the wind stress correction was associated with the angle offsets. The signed average error at each level showed a systematic dependence on measurement height (Table 3 and Fig. 6). On average, the measured wind stress was underestimated by 15% ± 2% (95% confidence interval) at 18.1 m, 13% ± 2% at 13.8 m, and 11% ± 2% at 8.7 m, and overestimated by 3% ± 3% at 3.5 m. Thus, the motion correction modified the wind stress profile, shifting wind stress to higher values at the upper levels and lower values at the near-surface level. The contribution of platform motion to the stress error was larger during high-wind periods (with more platform motion; Fig. 6a). The contribution of sensor-mounting offsets to wind stress error was largest at the lowest and highest levels (Fig. 6b).

Motion correction had a large effect on the crosswind stress −. The measured crosswind stress was generally negative and proportional to wind stress, and the corrected crosswind stress was close to zero, though the scatter remained large (Fig. 7). We calculated the wind stress “angle” using the along-wind and crosswind stress components, α ≡ tan⁻¹(−/−), which some studies have linked to interactions between the surface wave and wind fields (Rieder et al. 1994; Friehe et al. 2001; Grachev et al. 2003). The motion correction reduced the measured wind stress angle at all levels (Fig. 7, dashed curves compared to solid curves). The sensitivity of the stress angle to motion correction, even for the small amounts of motion on FLIP, suggests caution when calculating this quantity from measured data.

d. Contribution of linear versus angular motion to wind stress error

We analyzed the relative importance of the three terms of the motion correction in Eq. (4) to the along-wind stress: the effect of rotating the measured velocity vector (coordinate rotation term) 𝗧_ep𝗠_pau_a; the effect of platform linear velocity ∫𝗧_epp dt; and the effect of angular motions 𝗧_ep(Ω_p × r_p). Similar to previous studies (E98, Schulz et al. 2005), the angular contribution was negligible compared to the other two terms. In our data, this was due to the small angle rates and the relatively small distance (r_p) between the motion sensor and anemometers. For the lowest wind conditions and small waves, the linear platform motions contributed over 90% of the total motion correction to wind stress. In contrast, during the highest wind and wave conditions, the coordinate rotation term was larger, accounting for ∼75% of the stress correction. The shift of the motion correction from the linear to coordinate rotation terms resulted from increased pitching and rolling and more cross contamination of w by u and υ during the higher wind and seas states.

e. Motion correction of winds measured from R/P FLIP versus a conventional ship

We compared the effect of platform motion on wind and wind stress measured at the 8.7-m level from FLIP with in situ measurements from a conventional research ship, the R/V Wecoma. Results are shown for a 3-day overlap period when the average 10-m wind speed was 6.4 m s⁻¹, the average distance between Wecoma and FLIP was 10 km (maximum 42 km), and the relative wind direction for each platform was within ±60° of forward (Fig. 8). For the Wecoma data, the motion correction resulted in a sign change of the stress from negative (measured) to positive (corrected) for 56% of the periods during the overlap interval. The magnitude of the stress correction as a fraction of the corrected stress for Wecoma was 77% ± 22%, compared to only 4% ± 2% for FLIP during the overlap period. The stability of FLIP due to its 100 m of draft results in much less pitching and rolling than conventional ships. Vertical motion on FLIP is also highly damped, with platform vertical motions of order centimeters per second, compared to conventional ships where vertical motions are routinely of order meters per second, particularly at the ship’s bow, where wind turbulence sensors are typically mounted.