## 1. Introduction

Ship-mounted acoustic Doppler current profilers (ADCPs) can use attitude data to correct for errors due to the pitch, roll, and heading of the ship. Heading corrections (King and Cooper 1993) are well established. Unfortunately, pitch and roll data have not often been conveniently available or easy to interface to the processing hardware. This is certainly the case on ships run by the Natural Environment Research Council (NERC), on which the pitch and roll inputs to the RD Instruments (RDI) system employed have always been set to be zero. Only recently, with the advent of the global positioning system (GPS) instruments such as the Ashtech GPS 3DF, which has four antennas to measure the orientation of a plane in space, that pitch and roll information has been readily available. King and Cooper (1995) examined the corrections that can be made to the *heading* data derived from ship’s gyrocompasses. Here we examine the magnitude of errors generated by ignoring the pitch and roll components of the three-dimensional attitude rotation matrix. We have made the assumption that the ADCP transducers are misaligned only in terms of heading. In fact, there may also be pitch and roll misalignments that we have not considered here.

The motivation for this analysis comes from work on data collected in Drake Passage in 1994 aboard the RRS *James Clark Ross.* One of the aims of the work is to estimate the total geostrophic volume transport of water through the section. Thirty-one CTD stations were occupied across the section, and ADCP data were collected using an RDI 150-kHz system, both on station and underway. The CTD data is used to calculate the geostrophic shear between station pairs, while the ADCP provides a reference velocity. By using the ADCP reference velocity as a reference for the geostrophic shear profiles, estimates of the total geostrophic flow are obtained. The total geostrophic velocity profiles can then be used to determine the sectionwide geostrophic volume transport. The method attempts to overcome the old oceanographic problem of the unknown geostrophic reference level. The application of this method has many environmental aspects that are not appropriate to discuss here. We draw the readers attention to one recent paper (Saunders and King 1995). The authors discuss the environmental limitations to such a technique but show how novel and important aspects of ocean circulation may be illuminated by careful consideration of ADCP data. Our work is concerned with the technical limitations of ADCP data caused by motion of the platform on which the instrument is mounted.

The ADCP was operated in one of two modes: bottom track and water track. The bottom-track mode was used in shallow water (<500 m) and allowed estimates of water depth and speed over the ground to be made. By emitting a long pulse (ping) between a set of normal pulses, the instrument measures the Doppler shift of the reflection from the ocean floor and thus the ship’s speed. In water-track mode, the instrument produces estimates of the speed of scatterers within the water column at regular depths and thus the speed of the water with respect to the ship.

Heading misalignment and velocity amplitude calibrations were obtained by comparing the velocity of the ship over the ground, determined from the ADCP in bottom-tracking mode, and from GPS positions over the same period (Joyce 1989; Pollard and Read 1989). The vector difference of the two velocity estimates represents the misalignment of the transducers with respect to the axis of the ship and includes errors in the sound speed used to calculate the water velocities (RD Instruments 1989). Heading misalignment transforms some of the fore–aft speed of the ship into the port–starboard component. The standard error of the estimate of this misalignment angle is typically 0.1°. At a ship speed of 6 m s^{−1} (approximately 12 kt), this would produce errors of 0.01 m s^{−1} in the cross-ship speed. This error velocity, if it appeared as a bias, would lead to large errors in volume transport estimates. For example, on the Drake Passage section considered here (section area = 2.57 × 10^{9} m^{2}) the volume transport error is 26 × 10^{6} m^{3} s^{−1}, which can be compared to the total geostrophic transport relative to a zero reference velocity at the bottom of 140 × 10^{6} m^{3} s^{−1}. The heading error is the limiting factor in the accuracy of ADCP measurements for volume transport estimates. Here we show that neglect of pitch and roll biases lead to errors a factor of 2 smaller than heading errors. We derive equations that may be used to correct ADCP data in a postprocessing mode. The corrections obtained for this dataset are small, which we believe is due to a fortunate combination of pitch and roll frequencies and phases for our cruise. Theoretical results will be shown that demonstrate that the errors can be very much larger if pitch and roll variations are in closer phase.

To provide a reference velocity from the ADCP at the same position as the geostrophic velocities, either on station or underway, velocities can be averaged. On-station averages represent the average underway velocities only if the velocity varies linearly across the interval. On the Drake Passage section, use of these two velocity averages produce estimates for the total volume transport that differ by 50 × 10^{6} m^{3} s^{−1}. The difference that is contributed by the attitude of the platform on station and underway is the object of study in this paper.

Figure 1 shows the pitch and roll, averaged to 2 min to correspond to the averaging of the ADCP data, as a function of latitude across the section. CTD station positions are revealed as concentrations of points in vertical lines. It is apparent that there is a bias of up to 2° in roll between on station and underway data. When hove to on station, the ship is held with the bow into the wind and up to 5000 m of cable lowered over the starboard side. We need to assess the impact of this discrepancy on the observed ADCP data. The pitch data do not have a similar bias but a steady, positive value of the order of 0.75°. This is attributed to the large volume of cargo being carried to one of the British Antarctic Survey (BAS) Antarctic bases, which pushed the ship’s bows downward. On the return trip, the pitch offset was consistently nearer to zero. If we take a vertical axis pointing downward, the positive on-station bias indicates the sign convention for the roll data: positive angles correspond to the vertical axis tilting to port. A positive pitch on the other hand, corresponds to the vertical axis tilting backward (Fig. 2).

Kosro (1985) examined the effects of pitch and roll on ADCP data collected off the coast of California in 1981 and 1982. Kosro was primarily interested in the effect of pitch and roll on short timescale velocity data. On these timescales he was able to ignore the bias resulting from pitch and roll of the ship (less than 0.01 m s^{−1}) because noise from other sources was larger (0.01 to 0.02 m s^{−1}). For the Drake Passage dataset, biases of this magnitude cannot be ignored.

If changes in attitude of the ship underway and on station are important we have to ensure that the calibration is valid for all ship’s attitudes. Section 3 and the appendix derive equations relating the velocity of the water relative to the ship as calculated by the RDI software to the true water velocity. Separate calculations are made for the water-track data and the bottom-track data to consider the effect of ship’s attitude on calibration. In section 4 we apply the pitch and roll corrections to data gathered on an NERC ship. A calibration is derived both with and without an attitude correction. Results for pitch and roll corrections are discussed in the final section.

## 2. Data

WOCE section SR1, Burdwood Bank to Elephant Island, was occupied from 15 to 21 November 1994. Thirty-one full-depth CTD stations were occupied, and continuous ADCP and 3D GPS data were collected. Transducer orientation is *π*/4 to the ship’s centerline (Fig. 2). Several hours of calibration data were obtained in shallow water off the Falklands shelf. ADCP profiles were 64 bins deep each with a depth of 8 m with the first bin center located at a depth of 13 m and averaged over a period of 2 min. Only the 2-min ensemble averages were available after the cruise, so we make the assumption that velocities are steady over this period. Velocity variations during the 2-min interval that correlate with pitch and roll variation may of course create a larger bias than described here. In bottom-track mode, one in four pings was selected to be a bottom-track ping.

An Ashtech GPS 3DF receiver provided 80% coverage. Quay-side positions from the GPS receiver have an rms error of 24 m.

## 3. Derivation

### a. Water track data

*i*th transducer is where

**r**

_{i}is a unit vector in the transducer direction,

**k**is the unit vertical,

*c*is the speed of sound in sea water,

*t*is time, and

*f*

_{0}is the Doppler-transmitted frequency. The velocity

**U**is assumed independent of horizontal position beneath the ship. Define rotation matrices in three dimensions as where

**R**

_{x},

**R**

_{y}, and

**R**

_{z}are rotations about the

*x, y,*and

*z*axes, respectively. Here

*C*= cos

_{θ}*θ*;

*S*= sin

_{θ}*θ*;

*T*= tan

_{θ}*θ*;

*xyz*is a right-handed coordinate system with the

*x*direction along the fore–aft axis of the ship, the

*y*direction across the ship, and the

*z*direction vertically downward (Fig. 2).

A general rotation in three dimensions can be defined in 1 of 12 ways (Wertz 1978), and it is important to use the correct sequence of rotations when applying observations of attitude to ship’s data. This is dependent on the definitions of pitch, roll, and heading adopted by the Ashtech attitude-determining software. The Ashtech receiver used here generates an attitude matrix that is obtained by the following successive rotations: *ψ* about the *y* axis (pitch), followed by *θ* about the *x* axis (roll), followed by *ϕ* about the *z* axis (heading).

*z*-axis rotation until after we have derived expressions for the horizontal components. The rotation matrix corresponding to pitch and roll rotations is

**r**

^{′}

_{1}

**A**

*θ, ψ*

**r**

_{1}

*π*/2 −

*γ*) is the angle made to the vertical (Fig. 2), giving The other unit vectors are related to the first by where the following identities have been used The velocity components along the transducer directions are then where is small, and

*z*

_{0}=

*ctS*

_{γ}is the bin depth when pitch and roll are zero.

*U*in a Taylor series and truncating to a first order gives

**r**′ vector represents the error in ignoring the direction the transducer faces as the ship rolls and pitches (the directional part). Now if the ADCP software assumes zero pitch and roll, the transducer velocities are combined on this basis. This is equivalent to setting

*ψ*and

*θ*to zero in Eq. (8a)–(8c) to give Rearranging for the velocity components relative to the ship in the transducer directions,

Equations (10a)–(10c) represent the calculations that the ADCP has performed. To obtain the real velocities, we need to substitute the full expressions for the *ω*_{i} and solve for the **U**(*z*_{0}).

*u*component directed along the first transducer direction. Assuming the shear terms are small, we can obtain closed expressions for the true velocity components on the RRS

*James Clark Ross*(see the appendix for details).

These are just Eqs. (A12), (A10d), and (A11d) from the appendix, where *δ* is the misalignment angle of the transducers; (*u*_{s}, *υ*_{s}, *w*_{s}) is the velocity of the water relative to the ship when pitch, roll, and misalignment are ignored; and (*u, υ, w*) is the true water velocity relative to the ship.

### b. Heading misalignment and velocity amplitude calibration using bottom-track data

Bottom-track data allows a direct comparison of the speed over the ground as calculated by the ADCP and that obtained from GPS data. The angular difference of these two vectors is assumed to be caused by the misalignment of the transducers *δ* by some small amount away from the axis of the ship. A systematic bias in pitch and/or roll may influence estimates of this vector difference. Here we derive expressions for the calibration parameters that take account of the ship’s attitude.

*u*

_{g}and

*υ*

_{g}are velocities over the ground in the ship frame from GPS data and

*k*is an amplitude correction factor that allows for errors in the sound speed used by the ADCP.

*u, υ, k,*and

*δ.*Substitute (13a) and (13b) into (12a) and (12b) and then divide to eliminate

*k.*This gives an expression for

*δ*Then

*k*can be found as

The calibration equations of Pollard and Read (1989) are a special case of Eqs. (15) and (16) with *θ* = *ψ* = 0. These equations are useful when the pitch and roll of the ship are steady since they show how much influence the ship’s attitude has on the calibration. In normal circumstances, where the pitch and roll may vary, the data are first corrected for pitch and roll using Eq. (11) and then calibrated using Eqs. (15) and (16) with *ψ* = 0 = *θ.*

## 4. Application to Drake Passage data

### a. Directional error

In this section we examine the special case [represented by Eqs. (11a)–(11c)] in which the vertical gradients are assumed to be small. The available ADCP velocities are ensemble averages over 2-min intervals, so we consider two cases: 1) the average of the product of pitch and roll is assumed to equal the product of their 2-min means and 2) correlations between pitch and roll are allowed in a simple fashion.

At a ship’s speed of 6 m s^{−1} (∼12 kt) this produces a correction in the cross-ship speed on the order of 0.005 m s^{−1}. This is small compared to random errors in the data, but if it appears as a consistent bias across the section, it introduces a volume transport error of 13 × 10^{6} m^{3} s^{−1}, or approximately 10% of the total.

*ω*

_{1}and

*ω*

_{2}and phase difference

*α*Substituting these into (17) gives the horizontal velocity components

Here, we have ignored the misalignment angle correction *δ* and have averaged over the 2-min period (denoted by an overbar), assuming that 2*π*/*ω*_{1} and 2*π*/*ω*_{2} are much smaller. The most significant terms are those that modify the *u*_{s} term since they transform some of the forward speed of the ship into a cross-ship speed. If *ω*_{1} = *ω*_{2} and *α* = 0, that is, the pitch and roll vary at the same frequency and are in phase, the correlation term *ψ*′*θ*′ reinforces the contribution of the mean term. If *ω*_{1} = *ω*_{2} and *α* = *π,* the correlation term opposes the mean term. For example, taking *;afψ**ψ*′ = 2°, *;afθ**θ*′ = 2°, *ω*_{1} = *ω*_{2}, and *α* = 0 gives, for ships speed of 6 m s^{−1}, a contribution of order 0.05 m s^{−1} to the across-ship speed, which gives the same order of magnitude as the total volume transport across the section. The influence exerted by these correlation terms is thus crucially dependent on the sea state and the response of the ship.

Figure 3 shows a 10-min interval of 1-s frequency pitch and roll data that is typical of the cruise. A fast Fourier transform of this data reveals that both pitch and roll consist of single broad spectral peaks centered on frequencies 0.025 s^{−1} apart. For this dataset then, the correlation term in Eq. (19) is not significant.

### b. Gradient error

^{−1}for the

*u*and

*υ*components). Using small angle approximations in Eq. (A1), the gradient terms appear within expressions of the form The contribution from the gradient terms can thus seen to be negligible for the depths at which we have data.

### c. Calibration

ADCP data were calibrated with a 60-h section of ship’s track over the Falkland Islands shelf for which bottom-track velocities and GPS data were available. The calculations are best performed over periods of steady speed and attitude. Figure 4 shows the pitch and roll data for this part of the cruise as a function of time and represents a period at the beginning of the cruise when bottom tracking was possible. There are clear periods when the heading and pitch of the ship were steady; however, the roll shows relatively large excursions because of the varying exposure to the wind field around the Falkland Islands.

Without a pitch and roll correction [(15) and (16) with *ψ* = 0 = *θ*] the data give a calibration of *δ* = −2.14 ± 0.18 and *k* = 1.00 ± 0.01. Employing 3D GPS data to provide a pitch and roll correction in Eq. (11), the calibration values are unchanged. We can confirm this is the case using the 5-h portion of data, around 2880 min from Fig. 4. We then obtain the terms in Table 1 that were formed by averaging the parameters over this steady interval. For (15), the heading misalignment, the pitch and roll corrections amount to a 0.0012% change in the numerator and a 0.001% change in the denominator. For the velocity amplitude, scaling pitch and roll corrections change the calibration by less than 0.005%. The pitch and roll thus result in very small changes in the size of the numerator and denominator and produce negligible effect in this calibration.

As in section 4a, an analysis of the effect of correlations within the averaging interval can also be undertaken for the calibration terms. For the dataset considered here there was a negligible effect.

## 5. Conclusions

We have derived equations that enable us to correct ADCP data for the effect of the attitude of the transducers. The error is in two parts: the first is caused by a change in direction of the transducers, the second by a change in depth of the averaging bin used by each transducer. The latter produces changes in the measured velocities because of the vertical shear of the current. The former has the effect of rotating a fraction of one velocity component into one or both of the others.

Two factors are likely to create these errors: the normal pitch and roll of the ship about its mean state and the orientation of the ship away from its rest state to a new mean by moments due to uneven loading (i.e., leading to a bias in pitch and roll from zero). Figure 1 clearly shows the effect of loading since the on-station data is significantly different from the underway data. Figure 4 shows that pitch and roll variations may be themselves of large amplitude compared to the mean state.

For the dataset considered here, corrections resulting from attitude variation are small and over short time or space intervals are not significant compared to other noise sources in the data. For transport calculations, however, in which the velocities are averaged over long sections, this bias accumulates and can be a significant fraction of the total. For this dataset, they amount to about 10% of the transport through the section.

Correlations of pitch and roll have been shown to be negligible for this dataset. However, if the pitch and roll variations are at the same frequency, significant errors in transport could be produced, which increase with decreasing phase difference. This is a function of sea state and ship response.

For NERC ships, assuming vertical velocity gradients are small, data can be corrected from Ashtech GPS 3DF data by taking averages of the pitch and roll terms in Eq. (11) instead of using the pitch and roll averaged over the ensemble interval.

It should be emphasized that the above corrections are purely caused by the geometry of the transducers with respect to the water. Pitch and roll variations are also likely to create short timescale accelerations of the ship that add to the velocity of the water relative to the ship. These errors may average out over large intervals of space, but it is possible that they correlate with the sort of errors discussed in this paper.

The authors are grateful to the crew and officers of the RRS *James Clark Ross* without whom the dataset used here could not have been collected. David Smeed and Gwyn Griffiths provided many helpful suggestions on the text.

## REFERENCES

Joyce, T. M., 1989: On in situ “calibration” of shipboard ADCPs.

*J. Atmos. Oceanic Technol.,***6,**169–172.King, B. A., and E. B. Cooper, 1993: Comparison of ship’s heading determined from an array of GPS antennas with heading from conventional gyrocompass measurements.

*Deep-Sea Res.,***40,**2207–2216.Kosro, P. M., 1985: Shipboard acoustic current profiling during the coastal ocean dynamics experiment. Ph.D. thesis, Scripps Institution of Oceanography, University of California, 119 pp. [Available from University of California, La Jolla, CA 92043.].

Pollard, R. T., and J. F. Read, 1989: A method for calibrating ship-mounted acoustic Doppler profilers and the limitations of gyro compasses.

*J. Atmos. Oceanic Technol.,***6,**859–865.RD Instruments, 1989: Acoustic Doppler current profilers. Principles of operation: A practical primer. RD Instruments, 36 pp. [Available from RD Instruments, 9855 Business Park Avenue, San Diego, CA 92131-1101.].

Saunders, P. M., and B. A. King, 1995: Bottom currents derived from a shipboard ADCP on WOCE cruise A11 in the South Atlantic.

*J. Phys. Oceanogr.,***25,**329–347.Wertz, J. R., 1978:

*Spacecraft Attitude Determination and Control.*Kluwer Academic Press.

# Derivation of Pitch and Roll Equations

In the text we have an expression for the ADCP-calculated velocities in terms of the transducer velocities [Eq. (10)] and an expression relating the transducer velocities to in situ water speeds [Eq. (8)]. Here, we combine them and then invert the result to give the water velocity as a function of the ADCP-measured velocity.

*u*component directed along the first transducer direction using the identities

*James Clark Ross,*the transducers are mounted at

*π*/4 to the axes of the ship, so rotating the above vector by

*π*/4 to bring it into the fore–aft/port–starboard frame of reference But Ω is nominally

*π*/4 with some error

*δ,*so and Equation (A6c) gives Equations (A7) and (A6a) give and Equations (14a) and (12) give and Eliminate

*w*from (A8b) and (A9b) to give and expanding the right-hand side Therefore, so Similarly, eliminating

*υ*from (A8b) and (A9b) gives So Finally, substituting (A10d) and (A12d) into (A7) gives

*u*

*C*

_{ψ}

*C*

_{δ}

*u*

_{s}

*C*

_{ψ}

*S*

_{δ}

*υ*

_{s}

*S*

_{ψ}

*w*

_{s}

*u*≡ fore–aft,

*υ*≡ port–starboard,

*w*≡ vertical) to the ADCP-measured speeds relative to the ship (

*u*

_{s},

*υ*

_{s},

*w*

_{s}).

A comparison of the terms in the ADCP calibration for misalignment angle (15) and velocity amplitude (16) when pitch and roll are included. Data are derived by averaging a 5-h period of steady conditions on the Falklands Shelf.