## 1. Introduction

An inverted echo sounder (IES) is an ocean-bottom instrument that measures the time for a 10-kHz sound pulse to travel round-trip to the ocean surface and back (Watts and Rossby 1977; Chaplin and Watts 1984). In use since the mid-1970s, these instruments provide up to two-year-long hourly time series of acoustic travel time *τ*. The *τ* measurements can be used to estimate the depth of isothermal surfaces in the main thermocline, the geopotential height anomaly between two pressure levels, or other dynamic and descriptive quantities (Rossby 1969; Watts and Johns 1982; He et al. 1997). Historical hydrography can be used to determine the empirical relationship between these quantities for a given region. Coincident measurements from a conductivity–temperature–depth (CTD) probe or an expendable bathythermograph (XBT) have been used to calibrate these relationships from *τ* (usually just determining an additive constant); effectively, this calibration is required to determine the precise depth of each instrument.

Although some IESs in earlier experiments have been equipped with pressure gauges (PIES), the *τ* and bottom pressure *P* records were calibrated and used independently. In particular, Watts and Kontoyiannis (1990) used the pressure measurements to test the drift and accuracy of the pressure sensors, and Shay et al. (1995) and Watts et al. (1995) used the pressure measurements to study deep geostrophic flows. This paper presents a new method of using *P* measurements to calibrate the measured acoustic travel times *τ*. The pressure measurement provides the instrument depth with improved accuracy. The new calibration, therefore, provides improved accuracy for the dynamic variables that can be estimated from the *τ* measurement, as will be shown.

## 2. Simulating an IES using historical hydrographic data

Examples of the functional relationships between acoustic travel time and various standard dynamic variables may be found in Watts and Rossby (1977) (Sargasso Sea), Watts and Johns (1982) (Gulf Stream), Chiswell et al. (1986) (eastern equatorial Pacific), Hallock (1987) (Gulf Stream and Sargasso Sea), Trivers and Wimbush (1994) (North Atlantic), James and Wimbush (1995) (North Pacific and Kuroshio Current), Garzoli and Bianchi (1987) (Malvinas and Brazil Currents), Garzoli and Gordon (1996) (Benguela Current), and Chiswell (1994) (Hawaii).

These papers have used a number of different methods to represent the vertical integral of acoustic travel time. The common goal among them is to simulate (from historical hydrographic data) an IES that measures temporal variations while moored at a fixed (*x, y, z*) point [where *z* represents absolute height, not depth measured below the sea surface, whose height itself varies with (*x, y, t*)]. Common to all of these simulations is the assumption that temporal variations at one site due to mesoscale eddy variability may be simulated from the combined (*x, y, t*) variations among a set of hydrographic profiles for the region. The need then is, on the one hand, to select from a space–time region data that are limited enough to exclude variability that would occur only “away” from the desired (*x, y*) site and, on the other hand, to select enough data to include and represent the full range of variability that can occur at the site (Hallock 1987; Trivers and Wimbush 1990; James and Wimbush 1995).

*z*or

*P*), and in particular the specification of integration limits, also differs among the aforementioned authors. This paper will not review the different approaches. The common goal is to represent the round-trip vertical integral of acoustic travel time to a fixed height

*z*in the ocean. This is represented mathematically as

*ρ, g, c,*and

*P*are the density, gravity, sound speed, and pressure, respectively. The choice to integrate between constant pressure limits is motivated by the following argument. First, bear in mind that the ocean surface height

*η*may change with steric height and with atmospheric pressure change and ocean barotropic pressure change, but hydrography can determine only the steric height changes. An alternative choice for the

*τ*integral would be to integrate the hydrographic profiles with height

*z*between the surface

*η*and a fixed distance below the sea surface. However, because of these substantial variations in

*η,*this is not as good a representation of a fixed height. A typical range of variation in the absolute height is approximately 1 m. On the other hand, it has been observed in the deep ocean that the bottom pressures and the surface atmospheric pressures vary independently, each by about 0.3 db (Qian and Watts 1992), which corresponds to approximately 0.3 m of hydrostatic height changes. Hence, the

*τ*representation in Eq. (1) is preferable because it more nearly represents a constant absolute height than an integration between

*z*limits.

*g*depends upon latitude. Hence, for the

*same*profile of salinity and temperature occurring at

*different*latitudes, the (

*P, z*) relation would differ (by more than 1 db per 5° of latitude at 3500 db). The approach taken here is to select a target latitude,

*ϕ*

_{t}, and stretch or shrink the

*P*axis from that that it has at the observed latitude,

*ϕ*

_{o}, to match that of the target latitude. This is done by using the algorithm of Fofonoff and Millard (1983) to convert the pressures from the hydrographic cast into depths, where

*g*=

*g*

_{o}(

*ϕ*

_{o},

*z*) is calculated using the observed latitude

*ϕ*

_{o}. The algorithm of Fofonoff and Millard is then inverted, and the depths are converted back into pressures,

*P*

_{t}. For this second calculation, however,

*g*=

*g*

_{t}(

*ϕ*

_{t},

*z*) is determined using the target latitude

*ϕ*

_{t}. This procedure is repeated for all of the hydrographic casts to be used from the region. In this manner, the latitudinal dependence of the pressure range of integration is removed, leaving the integration range as close to constant as possible, and the simulation of the IES is at a fixed latitude. For example, if

*P*

_{sim}= 2000 db, we denote

*ρ*′ and

*c*′ designating profiles as a function of

*P*

_{t}. (Note that deeper bottom pressures than 2000 db are simulated later; this calibration method is applicable to the full depth of any IES.)

^{1}

*τ*

_{2000}. The geopotential height anomaly is determined from the same hydrography via

*δ*represents the specific volume anomaly as a function of

*P*=

*P*

_{t}. Figure 1 shows

^{100}

_{4000}

*τ*

_{2000}, calculated from Eqs. (2) and (1) using about 130 hydrographic casts from the Newfoundland Basin near 42°N. The relationship between these quantities could be represented by a polynomial; it is adequate for our purpose of demonstrating the improved accuracy of the pressure method of calibration to approximate this relationship as linear (shown in Fig. 1 as a solid line):

^{100}

_{4000}

*A*

*τ*

_{2000}

*B,*

*A*≈ −270 m

^{2}s

^{−3}and

*B*≈ 737 m

^{2}s

^{−2}. If an IES in the Newfoundland Basin region was moored at a depth of precisely 2000 db, the slope and intercept values derived from this simulation could be directly applied to the time series of

*τ*

_{2000}measurements to obtain a time series of

^{100}

_{4000}

## 3. Calibration of the IES

The traditional method of calibrating the IES assumes that the slope *A* in Eq. (3) is not dependent on the pressure at which *τ* is simulated, so long as *P*_{sim} is far below the main pycnocline. Rather, the depth dependence of *τ* is solely absorbed by changing the intercept *B* to *B*′. Under this assumption, all that is required to determine ^{100}_{4000}*B*′ for each IES site.

*B*′ is to use information from one or more coincident CTDs. Equation (3) is rearranged to give

*B*

^{100}

_{4000}

*A*

*τ*

_{meas}

*τ*

_{meas}is the travel time measured by the IES at its actual bottom pressure

*P*

_{ies}and

^{100}

_{4000}

*B*′ is valid at

*P*

_{ies}. Multiple CTD drops at the IES site during the period of the deployment allow for multiple estimates of

*B*′, and averaging these produces a “best” estimate for

*B*′ (Tracey et al. 1997).

^{100}

_{4000}

*B*′ and random scatter. The two sources of random scatter are 1) the errors in the

*τ*

_{meas}record after 40-h low-pass filtering [(rms scatter of hourly measurements)/

*ε*

_{a}= 0.045 m

^{2}s

^{−2}], which propagate through Eq. (3) to be errors in

^{100}

_{4000}

*ε*

_{b}= 0.42 m

^{2}s

^{−2}). Errors in the determination of

*B*′ result in biases for the time series of

^{100}

_{4000}

*B*′: 1) the error in the hourly

*τ*

_{meas}[1 ms (Chaplin and Watts 1984) that corresponds to

*ε*

_{c}= 0.27 m

^{2}s

^{−2}] at the time of the coincident CTD, which affects the determination of

*B*′ via Eq. (4); 2) the scatter introduced because

*τ*

_{meas}in Eq. (4) represents an integration over the full water column, while the integration of

^{100}

_{4000}

*ε*

_{d}= 0.14 m

^{2}s

^{−2}based on the observed variability in density structure below 4000 db for this region); and 3) due to the spatial offset between the CTD and the IES sites (0.27 m

^{2}s

^{−2}per kilometer distance between the CTD and IES sites based on the maximum observed change in geopotential height anomaly across the front). The spatial offset error introduces a random scatter due to the deflections of isotherms by internal waves during the several hours involved in the CTD measurement as well as a bias due to the ambient horizontal gradient of the main thermocline depth. Note that when multiple CTDs are taken at a site, ε

_{c}, ε

_{d}, and ε

_{e}are reduced by a factor of

*N*

*N*is the number of CTD casts. An estimate of the total rms error in this traditional method, denoted

*ε*

_{tm}, is given by

## 4. Calibrating a PIES

The inclusion of pressure sensors on the PIES provides an alternative calibration method. ParoScientific, Inc., the manufacturer of the pressure sensors used in the PIES, states in its technical brochures an absolute pressure accuracy of 0.01% of full scale, or 0.5 db, in up to 5000-m depth. The accuracy of the pressure sensors has also been tested by predicting the bottom pressure of the PIES using the measured travel times and a sound-speed profile based on coincident full-water-column CTDs. For 11 PIES in three different experiments the resulting mean offset between the predicted pressure and the measured pressure was about 1 db, about half of which may be attributable to errors in the sound-speed algorithm rather than the pressure sensor (Meinen and Watts 1997). Thus, the pressure sensor provides the accurate depth information required to calibrate an IES *τ* record. Calibration for an individual PIES using this new method consists only of using the historical hydrography to simulate IESs, where *P*_{sim} = *P*_{ies} in Eq. (1), and then the *A* and *B* in Eq. (3) will apply directly to the measured travel times (rather than a 2000-db simulation). Also note that this calibration requires that we account for three well-known constant offsets intrinsic to the measurements used in this study. A discussion of these offsets is given in appendix A.

For a deployment involving multiple PIES, the method described above has the weakness that it would require determining a set of coefficients *A* and *B* for each individual PIES since no two instruments would be at exactly the same pressure. If these PIES were being calibrated into not only ^{100}_{4000}

A better approach is for the travel-time measurements from different sites to be projected onto a common (deep) pressure level, *P*_{com}, using *τ*_{com} = *τ*_{meas} + *A, B*). This approach requires only a single set of coefficients for other dynamic variables to be estimated from the *τ*_{com} measurements. Using historical hydrography to simulate IESs at various pressures, it can be shown that *τ*_{sim1} at any pressure level, *P*_{1}, which is significantly below the thermocline, is linearly related to *τ*_{sim2} at any other deep pressure level, *P*_{2}. Figure 2 shows a number of examples. The slopes are very nearly but not exactly 1. By studying a number of these relations it has been determined that the slope and intercept of the linear relationships between *τ*_{sim} at different pressures are simple functions of pressure themselves, *P*) and *P*). Appendix B presents the details of the conversion of the measured *τ*_{meas} at *P*_{ies} into *τ*_{com} on a common pressure level, *P*_{com}, which we take to be 2000 db in this study. This allows the use of a single set of *A* and *B* coefficients in Eq. (3) derived from a simulation of IESs at *P*_{com}.

*τ*

_{meas}record after 40-h low-pass filtering [(rms scatter of hourly measurements)/

*ε*

_{a}= 0.045 m

^{2}s

^{−2}], which propagate through Eq. (3) to be errors in Δ

^{100}

_{4000}

*ε*

_{b}= 0.42 m

^{2}s

^{−2}). Biases in the final calibrated

^{100}

_{4000}

*ε*

_{f}= 0.19 m

^{2}s

^{−2}], which results in errors in the

_{A}and

_{B}; and 2) the error introduced in converting the measured

*τ*

_{meas}values into

*τ*

_{com}(

*ε*

_{g}= 0.27 m

^{2}s

^{−2}), which comes from the scatter in fitting the

_{A}and

_{B}versus pressure curves. An estimate of the total error using this method of calibration, denoted

*ε*

_{pm}, is given by

*ε*

_{pm}

*ε*

_{a}

^{2}

*ε*

_{b}

^{2}

*ε*

_{f}

^{2}

*ε*

_{g}

^{2}

^{1/2}

^{2}

^{−2}

*P*

_{com}(2000–5000 db).

## 5. Comparison of the two calibration methods

From August 1993 until June 1995, four PIES were deployed in a line across the North Atlantic Current near 42°N (Tracey et al. 1996). During the period of deployment, one to three full-water-column CTDs were taken at each PIES site. Each CTD was used to determine a *B*′ intercept [via Eq. (4)] for determining ^{100}_{4000}*τ*_{meas}, and the *τ*_{meas} records were calibrated in the traditional manner described in section 3. These same *τ*_{meas} records were also calibrated using the pressure method described above, and the results of the two calibration methods were compared.

Figure 3 shows the mean of the 22-month time series of ^{100}_{4000}^{100}_{4000}*τ*_{meas} time series and by the error due to the scatter in Eq. (3). Thus, these two error sources are not included in the error bars shown in Fig. 3. The remaining sources of error are all biases, so the error bars would be the same for an individual day as they are for the 22-month averages shown in Fig. 3.

The error bars indicate that the ^{100}_{4000}^{2} s^{−2}, which is about 4% of the total North Atlantic Current signal in this region.

## 6. Summary

The addition of a pressure sensor to the IES has permitted the development of a new method of calibrating a time series of measured acoustic travel times into other dynamic variables without the need for coincident XBTs or CTDs at the IES site during the deployment. Instead, the method relies on the combination of historical hydrography from the region and the measurement of the pressure sensor.

One advantage of calibrating the PIES using the pressure method is that there is a smaller error inherent in the calibration. During the North Atlantic Current experiment the pressure method had a standard deviation in ^{100}_{4000}^{2} s^{−2}, while the traditional method had a standard deviation of 0.65 m^{2} s^{−2}. The majority (0.42 m^{2} s^{−2}) of these errors are due to the rms scatter of the linear relationship between *τ* and ^{100}_{4000}^{100}_{4000}*τ*; however, the improvement would apply to both calibration methods equally. Hence, for the simple purpose of demonstrating the improved accuracy of the pressure method of calibration, a linear fit is adequate.

A second advantage to the pressure method is the elimination of the need for coincident CTDs or XBTs, which reduces the cost and logistical efforts during deployment and recovery. Of course, the bottom pressure record is valuable in its own right to measure the barotropic pressure field (particularly if leveled by combination with deep current measurements), as shown in Shay et al. (1995), Howden (1996), and Lindstrom et al. (1997).

## Acknowledgments

The authors would like to express their appreciation to Karen Tracey for her assistance in processing the PIES data and in the preparation of this manuscript. Thanks also to Dr. Allyn Clarke and his colleagues at the Bedford Institute of Oceanography and Dr. Peter Koltermann at the German Hydrographic Service for providing some of the CTD data used in this project. Richard Wearn at ParoScientific, Inc., provided much useful information about the pressure sensors. The reviewers provided a number of useful comments, and our thanks go out to them. Finally, our thanks go out to the crews of the R/V *Oceanus* and CSS *Hudson* for their help in deploying and recovering these instruments. This project was funded under NOAA Grant NA56GP0134 and ONR Contract N00014-92-J-4013.

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## APPENDIX A

### Inherent Offsets in the PIES

Both the pressure and the travel-time measurements made by the PIES are subject to known constant offsets that must be removed prior to calibrating the travel times, as discussed in this paper. The pressure sensors discussed here measure absolute pressure, consisting of the pressure of the ocean plus the pressure of the overlying atmosphere. To combine with the PIES measurement of the acoustic travel time to the sea surface, it is necessary to subtract the atmospheric pressure from the measured pressures. Variations of the mean regional atmospheric pressure from one year to the next are less than 0.1 db, so it is adequate to subtract the annual mean regional atmospheric pressure value rather than to account for the mean over the specific 22-month time period of the experiment. Another offset originates because the acoustic transducer on the PIES is located 0.58 m above the pressure sensor. For combination with the travel time, the corresponding 0.6-db hydrostatic offset was subtracted from the measured pressure. Finally, the IES echo detector has a 3-ms internal response delay in detecting the returning sound pulse. This travel-time delay must be subtracted from the measured travel times to avoid overestimating the depth, and therefore the pressure, of the IES. (This delay has no effect on the traditional method of calibration because all bias errors are combined into the *B*′ determined from the concurrent CTDs or XBTs.)

## APPENDIX B

### Projecting *τ* onto a Common Pressure Level

*τ*at any one pressure level,

*P*

_{1}, considerably below the thermocline is linearly related to

*τ*at any other deep pressure level,

*P*

_{2}:

*τ*

_{P1}=

*τ*

_{P2}+

*τ*are on the order of a few milliseconds, while the absolute value of

*τ*is typically a few seconds, errors are minimized in fitting these linear relationships if a large value is subtracted from

*τ*to avoid numerical error. We simply subtracted a constant for each depth, which is the round-trip travel time that would be measured at a given pressure if the ocean had a constant sound speed of 1500 m s

^{−1},

*τ*

_{ms}:

*τ*

*τ*

*τ*

_{ms}

*τ*

^{′}

_{P1}=

*τ*

^{′}

_{P2}+

*τ*′ at a number of depths have demonstrated that slopes

*P*), and an intercept,

*P*), can be determined using the pressure measured by each PIES,

*P*

_{ies}. Figure B1 shows the slopes and intercepts obtained for converting

*τ*′

_{meas}at depths between 2000 and 5000 db into

*τ*

^{′}

_{com}

*P*

_{com}. Equation (B3) becomes

*τ*

^{′}

_{Pcom}=

*τ*

^{′}

_{Pies}+

*P*

_{ies}) and

*P*

_{ies}). With each of the individual PIES,

*τ*′ records projected onto

*τ*

^{′}

_{Pcom}

*A*and

*B*of Eq. (3), to convert a

*τ*′ time series from any depth into a time series of

^{100}

_{4000}

*P*

_{com}, is arbitrary as long as the level chosen is significantly below the main thermocline.

Another interesting piece of information can be gleaned from Fig. B1. Note that *τ*′ simulated at two different pressure levels is not equal to 1, it can be seen that the historically used assumption that *A* [in Eq. (3)] was independent of depth could have lead to errors of several percent in ^{100}_{4000}

Comparison of the round-trip travel time *τ*_{sim} at 2000 db to *τ*_{sim} at 2500, 3500, and 4500 db as labeled on the *x* axis. Units are seconds.

Citation: Journal of Atmospheric and Oceanic Technology 15, 6; 10.1175/1520-0426(1998)015<1339:CIESEW>2.0.CO;2

Comparison of the round-trip travel time *τ*_{sim} at 2000 db to *τ*_{sim} at 2500, 3500, and 4500 db as labeled on the *x* axis. Units are seconds.

Citation: Journal of Atmospheric and Oceanic Technology 15, 6; 10.1175/1520-0426(1998)015<1339:CIESEW>2.0.CO;2

Comparison of the round-trip travel time *τ*_{sim} at 2000 db to *τ*_{sim} at 2500, 3500, and 4500 db as labeled on the *x* axis. Units are seconds.

Citation: Journal of Atmospheric and Oceanic Technology 15, 6; 10.1175/1520-0426(1998)015<1339:CIESEW>2.0.CO;2

Plot of the mean of the 22-month time series of ^{100}_{4000}

Plot of the mean of the 22-month time series of ^{100}_{4000}

Plot of the mean of the 22-month time series of ^{100}_{4000}

Fig. B1. Slope *P*_{1} = 2000 db and *P*_{2} along the *x* axis. Solid lines are cubic polynomial fits to (

Fig. B1. Slope *P*_{1} = 2000 db and *P*_{2} along the *x* axis. Solid lines are cubic polynomial fits to (

Fig. B1. Slope *P*_{1} = 2000 db and *P*_{2} along the *x* axis. Solid lines are cubic polynomial fits to (

^{1}

An alternative method, which we had adopted after this paper went to press, is to perform the *τ*_{sim} and *τ*_{2000} integrals from CTD data using a constant *g* = 9.8 m s^{−2}. Then to calibrate the PIES at a given latitude and depth, one must only accurately account for the ratio of its site-specific *g*(*ϕ*, *z*) to 9.8 m s^{−2}.