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Correcting for Beam Spread in Acoustic Doppler Current Profiler Measurements

SEPTEMBER 004 NOTES AND CORRESPONDENCE 1491 Correcting for Beam Spread in Acoustic Doppler Current Profiler Measurements R. F. MARSDEN Physics Department, Royal Military College of Canada, Kingston, Ontario,
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SEPTEMBER 004 NOTES AND CORRESPONDENCE 1491 Correcting for Beam Spread in Acoustic Doppler Current Profiler Measurements R. F. MARSDEN Physics Department, Royal Military College of Canada, Kingston, Ontario, Canada R. G. INGRAM Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada 31 October 00 and 3 March 004 ABSTRACT Spatial homogeneity assumptions inherent in the conversion of directly measured acoustic Doppler current profiler (ADCP) beam to Cartesian coordinates for the Janus configuration are investigated. These assumptions may be adequate for large-scale flows, such as tidal currents and wind-forced upwelling. However, for highfrequency features, such as internal solitons and turbulence, the velocity fields may vary over scales comparable to the divergence of the acoustic beams. Equations are derived for beam spreading, and it is shown that a firstorder correction can be applied to improve velocity measurement accuracy. Two cases are examined. First, the effects of the spatial and temporal convolution inherent in beam spreading from the Janus configuration ADCP are applied to a model internal solitary wave. It is shown that the corrected vertical velocities have deviations of less than mm s 1 for distances up to 30 m from the transducer face and are approximately 3 times more accurate than the uncorrected velocities for distance up to 0 m from the transducer face. Next, under a frozen turbulence hypothesis, the method is applied to processing turbulence data. It is demonstrated that the horizontal longitudinal velocity can be markedly improved. 1. Introduction The acoustic Doppler current profiler (ADCP) has taken an increasingly important role in the measurement of both first-order (mean) and second-order (eddy correlation) current statistics. First-order statistics have been used to identify both larger-scale barotropic and baroclinic tidal (Foreman and Freeland 1991; Marsden and Greenwood 1994) and mean flows. At second order, van Haren et al. (1994) and Stacey et al. (1999) have estimated vertical profiles of eddy correlations. Marsden et al. (1994a,b) mounted a narrowband instrument through land-fast ice in the Canadian Arctic and detected numerous instances of apparently significant vertical velocities. Marsden et al. (1995) exploited these vertical velocities to propose a technique for determining internal wave directional spectra. In both instances, the ensemble-averaged velocities were assumed to accurately represent the flow. Problems may arise, however, when measuring first-order small-scale phenomena such as internal waves, solitons, and turbulence. Corresponding author address: Dr. Richard F. Marsden, Physics Dept., Royal Military College of Canada, P.O. Box Station Forces, Kingston, ON K7K 7B4, Canada. The ADCP has either three or four transducers mounted in a symmetric configuration at predetermined angles (typically 0 or 30) about the vertical (see Fig. 1). It measures a velocity profile by emitting one or more acoustic pulses. The return echo is range gated, and the Doppler shift of the radial velocity is determined in each bin. Of particular interest, for this paper, is the Janus configuration, whereby four transducers are oriented in vertical planes aligned at 90 between adjacent beams. Lohrmann et al. (1990) have proposed a novel technique for estimating turbulence parameters (specifically Reynolds stress) using this beam orientation. Because of the diverging beams, however, assumptions must be made about the flow statistics to convert from radial to Cartesian velocities. First, the beams measure a spatial average that is rotated relative to the vertical, as is indicated in Fig. 1. Consequently, a portion of the beams overlap adjacent depth cells. RD Instruments (RDI 1996, p. 16) indicates a correction for this effect. More importantly, however, opposing transducers sample different portions of the water column, indicated by the distance d tan in Fig. 1. If the length scale of the feature is the same as the beam spread, then velocity errors can occur. In this note an approach will be presented to correct 004 American Meteorological Society 149 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 FIG. 1. Schematic of the ADCP beams showing the variables defined in the text. velocities for beam spread. The correction is based only on the geometry of the Janus configuration and is independent of ADCP manufacturer. The note will be organized in the following manner. Section will present a synopsis and analysis of the inherent convolutions in the data. In section 3, simulations for internal waves and turbulence measurements will be presented. Discussion and conclusions will appear in section 4.. Sampling theory and data Sampling theory The radial velocities (b 1, b, b 3, b 4 ) for a four-beam downward-looking convex head in the Janus configuration (Fig. 1) actually measure the following Cartesian velocities: b 1(x 0, y 0, z i) u[x0 s(d i), y 0, z i] sin w[x s(d ), y, z ] cos, 0 i 0 i b (x, y, z ) u[x s(d ), y, z ] sin 0 0 i 0 i 0 i w[x s(d ), y, z ] cos, 0 i 0 i b (x, y, z ) [x, y s(d ), z ] sin i w[x, y s(d ), z ] cos, b (x, y, z ) [x, y s(d ), z ] sin i w[x, y s(d ), z ] cos, (1a) (1b) (1c) (1d) where x 0 and y 0 represent the nominal horizontal position of the transducer and z i is the nominal depth for the ith cell; b j (x 0, y 0, z i ) represents a beam velocity, positive toward the transducer j 1 4; u,, and w are the Cartesian east, north, and vertical velocities, respectively; is the deviation of the axis of the beam from vertical; and s(d i ) d i tan is half the horizontal beam separation at a depth cell a vertical distance d i from the transducer face (see Fig. 1). A little algebra isolates the Cartesian velocities in a more convenient form: b1 b 1 {u[x0 s(d i), y 0, z i] sin u[x0 s(d i), y 0, z i]} 1 {w[x0 s(d i), y 0, z i] w[x0 s(d i), y 0, z i]} cot, b4 b 1 3 {[x 0, y0 s(d i), z i] sin [x 0, y0 s(d i), z i]} (a) 1 {w[x 0, y0 s(d i), z i] w[x 0, y0 s(d i), z i]} cot, b1 b b3 b 1 4 {w[x0 s(d i), y 0, z i] 4 cos 4 w[x0 s(d i), y 0, z i] w[x, y s(d ), z ] w[x, y s(d ), z ]} 1 {u[x0 s(d i), y 0, z i] 4 u[x0 s(d i), y 0, z i] [x, y s(d ), z ] (b) [x, y s(d ), z ]} tan. (c) If the horizontal dimensions of the feature (e.g., soliton or turbulence patch) are larger than the beam separation, then one can expand () in a Taylor series about x 0, y 0, z i : b1 b w ũ u(x 0, y 0, z i) di sin x 1 u di tan, (3a) x b4 b w 3 (x 0, y 0, z i) di sin y 1 di tan, (3b) x SEPTEMBER 004 NOTES AND CORRESPONDENCE 1493 b1 b b3 b 4 w w(x 0, y 0, z i) 4 cos 1 u di tan x y 1 w w i 4 x y d tan. (3c) The error consists of a term proportional to the distance from the transducer face times the horizontal shear of the perpendicular component at first order and the square of the distance times the second derivative of the normal component velocity at second order. If the horizontal gradients in the velocity field are small, corresponding to large spatial scales, only the zero-order terms are retained, and the transformations normally used to estimate ADCP mean velocities are obtained. Equations (3) represent the spatial convolution inherent in a single ADCP measurement. A temporal averaging may also be performed that further distorts a measurement of the true velocity field. If the water is incompressible (a good assumption near the surface), then u w. (4) x y z Substitution into Eq. (3c) yields a first-order correction for w given by 1 ŵ w ŵ(x 0, y 0, z 0) di tan, (5) z where ŵ is a corrected estimate of w. The correction is based on a first-order vertical derivative only, can be obtained readily from a single profile, and is independent of the horizontal orientation (azimuth) of the ADCP. For turbulence studies, time scales are generally faster than the buoyancy period and the fluctuations are assumed to be advected by the mean flow (e.g., Marsden et al. 1993; McPhee 1994), the frozen turbulence hypothesis. Typically, the coordinate frame is rotated along the mean (0 min to hourly averaged) flow and the time derivatives are converted to spatial derivatives. Alternatively, for propagating features, the axes may be rotated along the phase velocity. The downstream spatial derivatives can be converted to time derivatives through w w (c U), (6) t x where c and U are the phase speed and mean advective velocities, respectively. Two methods were considered for the implementation of the corrections. First, a numerical technique was used to invert Eq. (5). This method suffers from an incomplete specification of the initial boundary condition. For the second method, the vertical derivative was approximated from the raw data; that is, 1 w ŵ(x 0, y 0, z 0) w di tan. (7) x It was found that the two methods, except for the first bin, produced almost identical results and, given its ease of implementation, only (7) will be considered further. Corrections to the horizontal velocity were then made using the corrected vertical velocity as follows: 1 ŵ û(x 0, y 0, z i) ũ d i, (8) c U t where the primes indicate the frame rotated along the mean velocity. Note that the correction for the horizontal cross-stream velocity ( ) is assumed to be of second order. The absolute limit of ADCP resolution is, of course, governed by sampling theory. Features must have horizontal length scales greater that twice the bin size and separation to avoid aliasing. 3. Simulations a. Internal waves The first simulation mimics ensemble-averaging 100 pings in a -min ensemble from an ADCP with a 30 transducer angle. Modern pulse-to-pulse coherent ADCPs are quite accurate and for these system characteristics have errors of cm s 1 for the vertical and 0.4 cm s 1 for the horizontal velocities, respectively. The rms error in the vertical gradient is given by d w d tan (30)/(z), (9) and the overall error for the corrected vertical velocity is then ŵ w d. (10) Similarly, the error in the horizontal correction is given by t ŵ d/[(u c)t], (11) and the overall error is û ũ t. (1) Averaging would typically be used for a long-term deployment with a low-frequency ADCP, such as that deployed by Schott and Leaman (1991) to detect vertical convection in the Gulf of Lyons or by Melling et al. (001) for a long-term survey of the North Water Polynya. Here accuracy, data storage, and battery power considerations may necessitate ensemble averaging. The specific example will be internal solitary waves detected by Marsden et al. (1994b) in the Canadian Arctic. They show that the Benjamin (1966) model gives a crude, but fairly reasonable, estimate of some waves. It is a 1494 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 shallow water, Boussinesq model for which the streamlines are given by z a(z) sech (x), (13) with horizontal and vertical velocities ac(z) sech (x) (x, ), 1 a(z) sech (x) 3.0ac(z) sinh(x) sech (x) w(x, ), (14) 1 a(z) sech (x) where w and are the vertical and horizontal velocities; a and c are the amplitude and phase speed, respectively, and are negative for waves of depression and waves traveling in the negative direction; and is the eigenvector and its vertical derivative corresponding to the largest eigenvalue (c 0 ) of the Sturm Liouville problem: c0 g 0 0; z z z z 0, h. (15) The phase speed (c) and length scale () of the wave are given by 0 3 dz h 0 0 dz h 0 c dz h 0 3 c0 dz h c c 1 a, (16) a 4. (17) FIG.. Contour of simulated (a) vertical (w) and (b) anomaly horizontal (u) velocity fields. For (a) the dashed and solid lines represent downward and upward flows, and the contour interval is 0.0 m s 1. In (b) the contour interval is 0.05 m s 1. Profiles of the density,, and were based on a cast taken at 101 eastern standard time (EST) 30 April 199 outside of Resolute, Nunavut, Canada. Details of the cast can be found in Marsden et al. (1994a). Care must be taken in implementing the model. The correction to the vertical velocity field [Eqs. (4) and (5)] is critically dependent on the incompressibility assumption. The model velocities were applied at the streamline depth, the results were differentiated numerically, and the incompressibility condition (critical for this method) was verified. Equation (14) was then used to calculate a theoretical 15-m solitary wave that was then propagated in the x direction at the net speed of the wave (c U) 40 cm s 1, where c is the intrinsic phase speed from (14) and U is the background tidal flow. Normally, solitons are embedded in the mean flow. The time of the disturbance can be found from the vertical velocity. The background flow is then taken to be an average of the flow immediately preceding and following the flow disturbance. The model was then sampled and averaged at the appropriate depth-dependent separations (i.e., d i tan) as outlined in () to imitate the spatial convolution inherent in the measurement. These values were compared to an equivalently averaged direct wave (i.e., a simulated average of the depth cells at one point directly beneath the transducer face). The model horizontal and vertical velocities prior to convolution and averaging are shown in Figs. a and b. The maximum is about 9 cm s 1 and is highly surface trapped within about 0 m of the ice water interface, while the model predicts maximum vertical velocities of 7.7 cm s 1 at 63-m depth. The maximum absolute velocity differences between the direct and convolved wave were calculated to quantify the beam divergence problem. Alternatively, the rms errors could have been used, but these have the disadvantage of presenting an averaged view of the errors. The outer extremities of the disturbance will correspond SEPTEMBER 004 NOTES AND CORRESPONDENCE 1495 FIG. 3. (a) Maximum absolute error of the uncorrected (crosses) and corrected (stars) estimates of the vertical velocity as a function of depth. (b) The unconvolved (solid), convolved (dot dashed), and corrected (dashed line) waves as a function of time at 17-m depth. to low velocities and hence present a conservative estimate. The maximum absolute error for the ŵ (Fig. 3a) peaks at 17-m depth with a difference of about 1.3 cm s 1 (or about 0%) of the peak amplitude. The rms ŵ errors as a function of depth using (9) and (10) are indicated by the solid line. The corrected amplitudes are below instrument noise levels to 40-m depth, below which the correction does not improve the estimate. Figure 3b shows the time series of the direct, or unconvolved (solid line), the convolved (dot dashed line), and the corrected (dashed line) waves. The large discrepancies at the peaks are evident, as is the efficacy of the velocity correction. Figure 4a shows the expected maximum absolute error of the downstream velocity, u, as a function of depth. Here, a highly idealized correction was calculated using the 0.4 m s 1 phase speed (c U) in (8). The uncorrected errors are less than cm s 1 to 0 m from the transducer head and increase sharply below to more than 8 cm s 1 at 50-m depth. The FIG. 4. Same as Fig. 3 for the horizontal velocity at 33-m depth. solid line indicates expected instrument noise levels as determined from (11) and (1). The corrections are greater than the noise level below 30 m, and indeed errors in beam divergence dominate the signal. The correction reduces the maximum error throughout the water column and is less than the instrument noise level within 40 m of the transducer head. Figure 4b shows the true, convolved, and corrected velocities at 33-m depth. The error is quite profound and can be corrected to nearly match the wave profile. It must be noted, however, that the horizontal phase speed will probably never be known with absolute certainty from a single point measurement, and while the simulations demonstrate the potential gains in correcting the horizontal velocities, these may never be realizable. b. Turbulence simulations The previous results demonstrated the effects of beam divergence on internal solitary waves that have length scales that are relatively large compared with beam divergence. Several investigators (e.g., Stacey et al. 1999; 1496 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 FIG. 5. Same as Fig. for the simulated turbulence field. The contour intervals are 0.01 and 0.05 cm s 1 for the vertical and horizontal velocities, respectively. Lu and Lueck 1999) have extended the use of the ADCP to turbulence scales. Here, the length scale of the flow would be of the same size as beam divergence, even very close to the transducer head. Marsden (004, manuscript submitted to J. Phys. Oceanogr.) has investigated turbulence near the ice water interface in the Canadian Arctic. He used a 614-kHz RDI ADCP set to pulse-to-pulse coherent mode (mode 11), sampling at 1.0 s in 8 depth cells at 0.5-cm increments. The instrument head was positioned 0.54 m below the ice water interface yielding the first sampling bin at 0.94 m. This mode was very accurate as sampling during quiescent periods indicated rms errors of 0.15 and 1. cm s 1 per ping in the vertical and horizontal velocities, respectively. The increased resolution and accuracy came at the cost of penetration, as results could only be obtained within 8mofthetransducer head. The minimum horizontal resolvable length scales, that are not aliased, range from 0.5 m in bin 1, due to the 0.5-m bin size, to 10. m at 7 m from the transducer head, due to beam divergence. Simulations suggest that beam divergence is a problem and can be rectified in the vertical and longitudinal velocities. For turbulence studies, normally (McPhee 1994) the mean flow is determined over, say, a 1-h period, and turbulence parameters are calculated from the deviatory velocities. The form of the streamfunction for modeling turbulent fluctuations was taken to be [ ] z 1 x (x, z) az exp b. (18) b c The horizontal (u /z) and vertical (w / dx) velocities were calculated for a number of estimates of b, c, and background flows ( U). The values of b and.0 c represent the length scales of the vertical and horizontal velocities, respectively. This form forces the vertical velocity fluctuation to be 0.0 at the wall. The amplitude a was adjusted to give typical velocities found in the data. In all cases, beam divergence was significant for the horizontal velocities and estimates could be improved. Figures 5a and 5b show the vertical and horizontal velocities associated with typical values of b.0, c 5.5, and U 40 cm s 1. The vertical velocity field is a doublet with maximum values of 3.0 cm s 1, while the horizontal velocity is surface trapped with a maximum value of about 17.0 cm s 1. The simulations were found to be insensitive to the advection speed U. A large value is preferable, however, to dominate any possible intrinsic phase speed of the eddies and to reduce the correction error of (11). Since the time scale of the fluctuations is greater than the minimum buoyancy period of the water column, the influence of internal waves should be small. Figure 6a shows that maximum error in the vertical velocity is very small (1.5 mm s 1 ), considerably less than the instrument noise level. Figure 6b indicates the direct (solid line), measured (dot dashed line), and corrected (dashed line) values for the 14th bin, 4.0 m from the transducer face. The errors are small and the correction only marginally improves the results near the transducer head. Figure 7a shows the maximum absolute error of the horizontal velocity as a function of depth. Maximum values of the uncorrected velocities are.0 cm s 1, above the instrument noise level, between 1.5- and 6.0-m depth, indicated by the solid line. Figure 7b shows the actual, measured, and corrected values at 4.0- m depth. The extent of the beam divergence error and its subsequent correction are quite striking. The simulations were run for considerable ranges in values for b and c, and the results were similar. Errors in the vertical velocity were negligible, while corrections to the hor
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